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Generalized Bernoulli numbers
edit on github · preview edits · show history · long url · generating function sequence special values
Numbers
$q$
$n$
$k$ 
$B_{k,\chi}$
1
1:
equals: Bernoulli_numbers
comment: $\chi = 1$
2
1
0:
1/2
2
1
2:
-1/6
2
1
4:
7/30
2
1
6:
-31/42
2
1
8:
127/30
2
1
10:
-2555/66
2
1
12:
1414477/2730
2
1
14:
-57337/6
2
1
16:
118518239/510
2
1
18:
-5749691557/798
2
1
20:
91546277357/330
2
1
22:
-1792042792463/138
2
1
24:
1982765468311237/2730
2
1
26:
-286994504449393/6
2
1
28:
3187598676787461083/870
2
1
30:
-4625594554880206790555/14322
2
1
32:
16555640865486520478399/510
2
1
34:
-22142170099387402072897/6
2
1
36:
904185845619475242495834469891/1919190
2
1
38:
-402695297194783210267101289/6
2
1
40:
143531742398845896012634103722237/13530
3
1
0:
2/3
3
1
2:
-1/3
3
1
4:
13/15
3
1
6:
-121/21
3
1
8:
1093/15
3
1
10:
-49205/33
3
1
12:
61203943/1365
3
1
14:
-5580127/3
3
1
16:
25949996501/255
3
1
18:
-2832495743227/399
3
1
20:
101471818419863/165
3
1
22:
-4469253897850313/69
3
1
24:
11126033443528968583/1365
3
1
26:
-3623473373642099263/3
3
1
28:
90551915146257709805297/435
3
1
30:
-295654219044074800178128205/7161
3
1
32:
2380921255285895617650701141/255
3
1
34:
-7164761463168472585088876287/3
3
1
36:
658296847749509543828094473602326169/959595
3
1
38:
-659664551880713792858389677483679/3
3
1
40:
529026058103493838442148858699253120783/6765
3
2
1:
-1/3
3
2
3:
2/3
3
2
5:
-10/3
3
2
7:
98/3
3
2
9:
-1618/3
3
2
11:
40634/3
3
2
13:
-1445626/3
3
2
15:
69214610/3
3
2
17:
-4291996834/3
3
2
19:
334634533226/3
3
2
21:
-32040806644042/3
3
2
23:
3696041900719682/3
3
2
25:
-505557954433400050/3
3
2
27:
80907883885186609178/3
3
2
29:
-14977166123084102901658/3
3
2
31:
3175377541259449430748914/3
3
2
33:
-764437634383264654750009666/3
3
2
35:
207382351212981234577556198090/3
3
2
37:
-62973639199791072683378619408874/3
3
2
39:
21275989531016261810482371942527906/3
4
1
0:
1/2
4
1
2:
-1/6
4
1
4:
7/30
4
1
6:
-31/42
4
1
8:
127/30
4
1
10:
-2555/66
4
1
12:
1414477/2730
4
1
14:
-57337/6
4
1
16:
118518239/510
4
1
18:
-5749691557/798
4
1
20:
91546277357/330
4
1
22:
-1792042792463/138
4
1
24:
1982765468311237/2730
4
1
26:
-286994504449393/6
4
1
28:
3187598676787461083/870
4
1
30:
-4625594554880206790555/14322
4
1
32:
16555640865486520478399/510
4
1
34:
-22142170099387402072897/6
4
1
36:
904185845619475242495834469891/1919190
4
1
38:
-402695297194783210267101289/6
4
1
40:
143531742398845896012634103722237/13530
4
3
1:
-1/2
4
3
3:
3/2
4
3
5:
-25/2
4
3
7:
427/2
4
3
9:
-12465/2
4
3
11:
555731/2
4
3
13:
-35135945/2
4
3
15:
2990414715/2
4
3
17:
-329655706465/2
4
3
19:
45692713833379/2
4
3
21:
-7777794952988025/2
4
3
23:
1595024111042171723/2
4
3
25:
-387863354088927172625/2
4
3
27:
110350957750914345093747/2
4
3
29:
-36315529600705266098580265/2
4
3
31:
13687860690719716241164167451/2
4
3
33:
-5858139922124796551409938058945/2
4
3
35:
2825315473256076432175888660864835/2
4
3
37:
-1525216232562154978526841947836584665/2
4
3
39:
916093640554681221828695314471758961899/2
5
1
0:
4/5
5
1
2:
-2/3
5
1
4:
62/15
5
1
6:
-1562/21
5
1
8:
39062/15
5
1
10:
-4882810/33
5
1
12:
16870116842/1365
5
1
14:
-4272460934/3
5
1
16:
55191040037254/255
5
1
18:
-16733932495095254/399
5
1
20:
1665220260620029882/165
5
1
22:
-203731775283813049306/69
5
1
24:
1408839768171310306622642/1365
5
1
26:
-1274511665105819697871886/3
5
1
28:
88473636765033006656216089798/435
5
1
30:
-802412747964728623624308412369810/7161
5
1
32:
17949661801608745008703191848268454/255
5
1
34:
-150041180800588335841892860936344254/3
5
1
36:
38293760085148791219398831829518230431269126/959595
5
1
38:
-106592557634632939880248157705153799915158/3
5
1
40:
237453349196993369379015348255478109059270556162/6765
5
2
1:
-0.600000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 + i * -0.200000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
5
2
3:
2.400000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 + i * 1.200000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
5
2
5:
-29.60000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 + i * -17.20000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
5
2
7:
778.4000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 + i * 473.2000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
5
2
9:
-35373.60000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 + i * -21769.20000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
5
2
11:
2.461914400000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000e6 + i * 1.519905200000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000e6
5
2
13:
-2.431537576000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000e8 + i * -1.502364812000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000e8
5
2
15:
3.233375441040000000000000000000000000000000000000000000000000000000000000000000000000000000000000000e10 + i * 1.998199869720000000000000000000000000000000000000000000000000000000000000000000000000000000000000000e10
5
2
17:
-5.569281693581600000000000000000000000000000000000000000000000000000000000000000000000000000000000000e12 + i * -3.441946719353200000000000000000000000000000000000000000000000000000000000000000000000000000000000000e12
5
2
19:
1.206157520505466400000000000000000000000000000000000000000000000000000000000000000000000000000000000e15 + i * 7.454431655968492000000000000000000000000000000000000000000000000000000000000000000000000000000000000e14
5
2
21:
-3.207991490791620456000000000000000000000000000000000000000000000000000000000000000000000000000000000e17 + i * -1.982645663375151852000000000000000000000000000000000000000000000000000000000000000000000000000000000e17
5
2
23:
1.027930850670510734824000000000000000000000000000000000000000000000000000000000000000000000000000000e20 + i * 6.352960344694664396120000000000000000000000000000000000000000000000000000000000000000000000000000000e19
5
2
25:
-3.905668677674515773514960000000000000000000000000000000000000000000000000000000000000000000000000000e22 + i * -2.413835830747005923837720000000000000000000000000000000000000000000000000000000000000000000000000000e22
5
2
27:
1.736252071531454020394805840000000000000000000000000000000000000000000000000000000000000000000000000e25 + i * 1.073062775366935853363923320000000000000000000000000000000000000000000000000000000000000000000000000e25
5
2
29:
-8.927895054861016602769457693600000000000000000000000000000000000000000000000000000000000000000000000e27 + i * -5.517742568914990825403472929200000000000000000000000000000000000000000000000000000000000000000000000e27
5
2
31:
5.257899694120307308164962949994400000000000000000000000000000000000000000000000000000000000000000000e30 + i * 3.249560717020431283356530051465200000000000000000000000000000000000000000000000000000000000000000000e30
5
2
33:
-3.516061694415380375981310934643277600000000000000000000000000000000000000000000000000000000000000000e33 + i * -2.173045633124581186392705355972441200000000000000000000000000000000000000000000000000000000000000000e33
5
2
35:
2.649620773849545198812482648609231290400000000000000000000000000000000000000000000000000000000000000e36 + i * 1.637555695430248488176264599727919057200000000000000000000000000000000000000000000000000000000000000e36
5
2
37:
-2.234952085776125730923511253831011474301600000000000000000000000000000000000000000000000000000000000e39 + i * -1.381276352214643397539751304501292660113200000000000000000000000000000000000000000000000000000000000e39
5
2
39:
2.097474513988285672624234862508522822075146400000000000000000000000000000000000000000000000000000000e42 + i * 1.296310540176154719836711648453400906250009200000000000000000000000000000000000000000000000000000000e42
5
3
1:
-0.600000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 + i * 0.200000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
5
3
3:
2.400000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 + i * -1.200000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
5
3
5:
-29.60000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 + i * 17.20000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
5
3
7:
778.4000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 + i * -473.2000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
5
3
9:
-35373.60000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 + i * 21769.20000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
5
3
11:
2.461914400000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000e6 + i * -1.519905200000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000e6
5
3
13:
-2.431537576000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000e8 + i * 1.502364812000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000e8
5
3
15:
3.233375441040000000000000000000000000000000000000000000000000000000000000000000000000000000000000000e10 + i * -1.998199869720000000000000000000000000000000000000000000000000000000000000000000000000000000000000000e10
5
3
17:
-5.569281693581600000000000000000000000000000000000000000000000000000000000000000000000000000000000000e12 + i * 3.441946719353200000000000000000000000000000000000000000000000000000000000000000000000000000000000000e12
5
3
19:
1.206157520505466400000000000000000000000000000000000000000000000000000000000000000000000000000000000e15 + i * -7.454431655968492000000000000000000000000000000000000000000000000000000000000000000000000000000000000e14
5
3
21:
-3.207991490791620456000000000000000000000000000000000000000000000000000000000000000000000000000000000e17 + i * 1.982645663375151852000000000000000000000000000000000000000000000000000000000000000000000000000000000e17
5
3
23:
1.027930850670510734824000000000000000000000000000000000000000000000000000000000000000000000000000000e20 + i * -6.352960344694664396120000000000000000000000000000000000000000000000000000000000000000000000000000000e19
5
3
25:
-3.905668677674515773514960000000000000000000000000000000000000000000000000000000000000000000000000000e22 + i * 2.413835830747005923837720000000000000000000000000000000000000000000000000000000000000000000000000000e22
5
3
27:
1.736252071531454020394805840000000000000000000000000000000000000000000000000000000000000000000000000e25 + i * -1.073062775366935853363923320000000000000000000000000000000000000000000000000000000000000000000000000e25
5
3
29:
-8.927895054861016602769457693600000000000000000000000000000000000000000000000000000000000000000000000e27 + i * 5.517742568914990825403472929200000000000000000000000000000000000000000000000000000000000000000000000e27
5
3
31:
5.257899694120307308164962949994400000000000000000000000000000000000000000000000000000000000000000000e30 + i * -3.249560717020431283356530051465200000000000000000000000000000000000000000000000000000000000000000000e30
5
3
33:
-3.516061694415380375981310934643277600000000000000000000000000000000000000000000000000000000000000000e33 + i * 2.173045633124581186392705355972441200000000000000000000000000000000000000000000000000000000000000000e33
5
3
35:
2.649620773849545198812482648609231290400000000000000000000000000000000000000000000000000000000000000e36 + i * -1.637555695430248488176264599727919057200000000000000000000000000000000000000000000000000000000000000e36
5
3
37:
-2.234952085776125730923511253831011474301600000000000000000000000000000000000000000000000000000000000e39 + i * 1.381276352214643397539751304501292660113200000000000000000000000000000000000000000000000000000000000e39
5
3
39:
2.097474513988285672624234862508522822075146400000000000000000000000000000000000000000000000000000000e42 + i * -1.296310540176154719836711648453400906250009200000000000000000000000000000000000000000000000000000000e42
5
4
0:
0
5
4
2:
4/5
5
4
4:
-8
5
4
6:
804/5
5
4
8:
-5776
5
4
10:
1651004/5
5
4
12:
-27622104
5
4
14:
15920571604/5
5
4
16:
-483949589792
5
4
18:
468896302250604/5
5
4
20:
-22566900581985640
5
4
22:
33011440685167168004/5
5
4
24:
-2307883601079395222448
5
4
26:
4749824393928895324795804/5
5
4
28:
-454788654136206663070849016
5
4
30:
1252792519982873734870069598004/5
5
4
32:
-157398682128777897627929287150144
5
4
34:
559170466092323651066774174989030604/5
5
4
36:
-89232906238536824499663660495205546632
5
4
38:
397247007941448722078502736760647883541604/5
5
4
40:
-78486597234073386729626964065725586449795280
6
1
0:
1/3
6
1
2:
1/3
6
1
4:
-91/15
6
1
6:
3751/21
6
1
8:
-138811/15
6
1
10:
25143755/33
6
1
12:
-125284471321/1365
6
1
14:
45706820257/3
6
1
16:
-850303535348267/255
6
1
18:
371258049560506117/399
6
1
20:
-53200355263894712681/165
6
1
22:
9372700281130681758263/69
6
1
24:
-93331922026621210558133881/1365
6
1
26:
121583587296211038415484353/3
6
1
28:
-12153672226427582366292575899919/435
6
1
30:
158728149919185987123774651693144755/7161
6
1
32:
-5112989460521173148653845258381741227/255
6
1
34:
61544832328734635119290129227020943617/3
6
1
36:
-22618907456493980678502845737134515169437426023/959595
6
1
38:
90663605652401488518932150975666054442899809/3
6
1
40:
-290835151140117606831053233979416059575082879713521/6765
6
5
1:
-2/3
6
5
3:
10/3
6
5
5:
-170/3
6
5
7:
6370/3
6
5
9:
-415826/3
6
5
11:
41649850/3
6
5
13:
-5922729722/3
6
5
15:
1134081384850/3
6
5
17:
-281284596509858/3
6
5
19:
87722769712529770/3
6
5
21:
-33597252908389628234/3
6
5
23:
15502327024398065811010/3
6
5
25:
-8481855507605264686660850/3
6
5
27:
5429636257086663655134162970/3
6
5
29:
-4020402432814999411841591287706/3
6
5
31:
3409535676215934030267550035128050/3
6
5
33:
-3283234640272164456247287225903892802/3
6
5
35:
3562801665116743683902214220172146856650/3
6
5
37:
-4327515534034273914525713203031293534364138/3
6
5
39:
5848299470469482205223530660381064633533707170/3
7
1
0:
6/7
7
1
2:
-1
7
1
4:
57/5
7
1
6:
-2801/7
7
1
8:
137257/5
7
1
10:
-33628005/11
7
1
12:
227722129787/455
7
1
14:
-113037178807
7
1
16:
2861988330243369/85
7
1
18:
-1700800459512794267/133
7
1
20:
331728747868864949627/55
7
1
22:
-79547450325899877636513/23
7
1
24:
1078164847807815609603979187/455
7
1
26:
-1911716339008952864163083503
7
1
28:
260105531512713952813270235151453/145
7
1
30:
-4623699485963545060678819437076574005/2387
7
1
32:
202723512084547905004921256341733116169/85
7
1
34:
-3321348107400234959534901386350705999167
7
1
36:
1661452792723927962759139630717702947677029135061/319865
7
1
38:
-9064481112767166197431436556104082555842408359
7
1
40:
39577689433261483512570835489772218702290589874611707/2255
7
2
0:
0
7
2
2:
1.428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428 + i * -0.494871659305393512436413240430249247697944358231537322301659136986266576259657153451756053359213878
7
2
4:
-24.57142857142857142857142857142857142857142857142857142857142857142857142857142857142857142857142857 + i * 10.88717650471865727360109128946548344935477588109382109063650101369786467771245737593863317390270533
7
2
6:
930 + i * -446.8691083527703417300811561085150706712437554830782020383982006985987183624704095669357161833701325
7
2
8:
-64780.57142857142857142857142857142857142857142857142857142857142857142857142857142857142857142857142 + i * 31851.91947953234803445730180705256257883049067321466821262398869298406191437657302476882661841244209
7
2
10:
7.238921428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428e6 + i * -3.581042262589915601011179313791614782362873249192664683058107521269405757120638357808709587703546186e6
7
2
12:
-1186077468 + i * 5.876609990608046876667944319014980072764352218463487010203707480007756962311111198579225162706199149e8
7
2
14:
2.679335545814285714285714285714285714285714285714285714285714285714285714285714285714285714285714285e11 + i * -1.328043841806932307938071256436329863765659271278708527823920861374593695802284878690562497468894330e11
7
2
16:
-7.981346627557257142857142857142857142857142857142857142857142857142857142857142857142857142857142857e13 + i * 3.956439869901551282318728125485400982997701226174828072876371835056989392766580316502172119637922062e13
7
2
18:
30313375693869510 + i * -1.502704185846872741177658939939439209091276599807829664524528392177595419063959280951038270174029920e16
7
2
20:
-1.429730989822974800857142857142857142857142857142857142857142857142857142857142857142857142857142857e19 + i * 7.087551486507104366807564718051886149624404935184507401481408271646351701221206887068869695807296877e18
7
2
22:
8.198467354011195647021428571428571428571428571428571428571428571428571428571428571428571428571428571e21 + i * -4.064201609479443462344453993303434351700034589408295326564001071319804402839356391425830309379500671e21
7
2
24:
-5617047522583541061569976 + i * 2.784523189336778407700217344162874015089306707576237371396638675660567177057970647811954500824632237e24
7
2
26:
4.531665021457023461119635001428571428571428571428571428571428571428571428571428571428571428571428571e27 + i * -2.246469789575916065488254857937137781710757816892755601657198308178006163237042104616629507530471874e27
7
2
28:
-4.252222084225511729354098002960571428571428571428571428571428571428571428571428571428571428571428571e30 + i * 2.107942365086108605628814637301126572332612972603821676949607315274303856976825104240164977224596798e30
7
2
30:
4591679162599800711017357297923050 + i * -2.276220503271842417485862980508831884845420006338172648517348789061934569979812301175392284254230373e33
7
2
32:
-5.653528036089197003024043527857351876571428571428571428571428571428571428571428571428571428571428571e36 + i * 2.802607933922814244252960119273768582097554638905770109191894907239308050696188695268000162957413196e36
7
2
34:
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8
17:
1.879048524950814729587699723631154584373459849914803631785222299558577240361540336804932265513471768e18 + i * -2.240667990241647864513109994883146830580150858078072668597535567610486507839033865861557996216384651e18
11
8
19:
-1.969669345419182444132048532699979307746054966352517217773995152919255748973785205713800681777221796e21 + i * 2.348702642951448157802403888228141229392225674794499071257268634023278376702692653819653770880943230e21
11
8
21:
2.535532297474444363313059715037800879077321241306704113241582422998900382724904373162552183468665892e24 + i * -3.023449068296690712333462895009324639766747372787993572813394377455193760670090145325746054217670623e24
11
8
23:
-3.932289843494899729243080207185355227367730250010111087539488289815158259915338658648160355727431653e27 + i * 4.688983792919309541254613970625258322665264117711796161404858626087974120219015176093450190801141164e27
11
8
25:
7.231401276935290256236724857175405246891918618585912706782826336674816828566246786805084137944594411e30 + i * -8.622944615175624023353901746419340177956853698658572113359802506572445138308784974219732187968128510e30
11
8
27:
-1.555912684330928695590410292057332703301354328079685618194430097237388745083603866484256562218981131e34 + i * 1.855317912800920818718035637008939857747349185719172645224391031961120970932349940053762582628565150e34
11
8
29:
3.872281199963100812525801301591800803228878068012022784430062264026851211824132509870182274957373876e37 + i * -4.617426587299329957975130571843849740447728974254984732516118877366442644361547670688378911114491363e37
11
8
31:
-1.103762082556758912063112255257234526518068529408393107350838086644556767908761648743030743569030486e41 + i * 1.316159676716151838127897513856726699974263647449994743818894930460944121277491521444692566542706254e41
11
8
33:
3.572440652137136495205544737945105615554980614742093027836745516194538115993828553538381305684635699e44 + i * -4.259887529150251102833567198618867458482577086075836055495741201120205175339362662445941995434640995e44
11
8
35:
-1.302979604504874452504374039247875747476676794219783751393982873768022161784916800306655499326994808e48 + i * 1.553712743614317691794602323451705120619674328878129813060257484137484728031329597024753806563153774e48
11
8
37:
5.319459126464467281069070730270033814831625419681109813551211124362584485563149832742774334542220437e51 + i * -6.343085805027219321218388348531208575646064588891904884292558307593874623932797205932296003119672044e51
11
8
39:
-2.416246920135133884641705999603618769659224620580419229408685694888639638750188227696846536114210938e55 + i * 2.881206749787539108658369541909232050315929995547835507643589264294743824394813810099831169846652141e55
11
9
0:
0
11
9
2:
2.320714913181856418834695696594313411738069389161540777405291409925359831702217291015561016658403222 + i * -1.974118016852780334270950688942982144871749779676500505937544063055991703688908635814039971315662101
11
9
4:
-85.44976159363766369492510822966120145363981377694693188632536248291677973242084243998362002093116411 + i * 97.52437467451935734570800203909407282556965817829238856848518599230259281731545016043767952779414125
11
9
6:
7807.211297971470432350759626042462813529882349382728845838433453376075464489747439261007374072374746 + i * -9567.938410530981016216349859639284210906350737642000807860547234461280630451424556253818101443411638
11
9
8:
-1.339157524816659348630305266626201681252729196710113277135449142125659871320852156977322228185344808e6 + i * 1.667888774542965837507026160189650081844599496306560417995858890948205619480382112819256169387631466e6
11
9
10:
3.694131619482769970429431943563481773736074862721993513701669737159683334669581850883586456324512918e8 + i * -4.618357434738402357685852880507502671256853171445497240218701036322061566829143910363033970870575879e8
11
9
12:
-1.494603030372525816851442767011689985035617865494425715908019080955525561312774041065591819992421862e11 + i * 1.870237166552872936839785633775290486182054673020526789407980736945761559560449842910009048987819333e11
11
9
14:
8.337342505963895016388199765612144850855687106691086323339521148866512927102345802764067465219151722e13 + i * -1.043507810295163332917204841889164070955503164091549325199698746804827659479351465800332463255248691e14
11
9
16:
-6.132899610866179424117411227696137532496085130923093964401531619044912842231783680330870930760047510e16 + i * 7.676407737943743851740042132939479510443975608639831792575832151567247862516982096720830930563674943e16
11
9
18:
5.751925690920470428835230013771147055765071794798471654460348380749489289256865914029501201709332670e19 + i * -7.199651145071722653689267699338854212828631114185082311978767472458896943599330740442018413034917462e19
11
9
20:
-6.699194493194078647344004282180987453605029214012133978034685253179282402926092705524884526841498924e22 + i * 8.385370819478362334446981773318735309803355483791627207841773631373595642637101383939665003778429972e22
11
9
22:
9.486155037910101579573442765319335827740792265904338145683928145428440427801723354924292333045850665e25 + i * -1.187381502968892215740740936617003387554406135893108407429703886382472234379224751054701725207692952e26
11
9
24:
-1.604925713887657058995718457376405837340346906185508717592895849896604555410248290765506528219015133e29 + i * 2.008885064098016300231180997113202194052674955981593220054929589752790611111743795442623443514891857e29
11
9
26:
3.197377591858617719514372534420104160290875891026145418603261145706474189641843689566180781165824621e32 + i * -4.002156846960782157617935681958847324390941660597337444671437891577591964257136180381383632043208527e32
11
9
28:
-7.408688862949201954032767400664823337898913797903169433862382829796036385257751417236135185185259529e35 + i * 9.273454387990816583728257762555374223568703898213524400879166635265879512922238173235312501854982687e35
11
9
30:
1.975541888680534917912369639253402787012218884335378829770890807212020323706184033830787527173824766e39 + i * -2.472785400607045422962254257785625168531969336702182862683862215489966059252141266066181562210879725e39
11
9
32:
-6.006528589131342165970312841488677260559876495674455835465718765783571125304692866255157867757200272e42 + i * 7.518370681554055204741674624428360045192726527759768098366755647334919486023188918867074582064980245e42
11
9
34:
2.065580091128268008459005956333395025521159412175790694089233233209045571867493266432709105261542803e46 + i * -2.585486203946479586119605159912988704680553700822255371062072756729993295471359301902507905713924781e46
11
9
36:
-7.976974757459987716511255035770275939822838107981605845847557177886365037470933155162332156980382930e49 + i * 9.984777774684033912205453318350150286772504315436856797582439895554375614369187889543267977587479460e49
11
9
38:
3.437551173375637722175246461150937023920672737888342452616551763974222359063002029717865195222044528e53 + i * -4.302782144773982438730460120880838078592294194155285543993761967989108077878535925437939128463480496e53
11
9
40:
-1.643612380793337046245764242566588108423659083147956638035617570756239575207373294770287604972458187e57 + i * 2.057309301983584277773689314866322276246723768521105716319182638577871200148597369369108042413726271e57
11
10
1:
-1
11
10
3:
18
11
10
5:
-12750/11
11
10
7:
152082
11
10
9:
-33743250
11
10
11:
11392546506
11
10
13:
-5449138537050
11
10
15:
38583798266778390/11
11
10
17:
-2924265117969948450
11
10
19:
3065285744885757445434
11
10
21:
-3945903957928886905334250
11
10
23:
6119596903016438430475002738
11
10
25:
-123791955711645506830680862563750/11
11
10
27:
24213774547673962192034742731813802
11
10
29:
-60262085831150042245568758121074256250
11
10
31:
171772146391807172082653785745679724525026
11
10
33:
-555958397446060369659180673860225365913706050
11
10
35:
22305274619480471069282483750352928827339774938910/11
11
10
37:
-8278368373500591521458576823199583266490145806557450
11
10
39:
37602661493633999417519944598054334637161263471575273554
Definition
Let $\chi$ be a Dirichlet character of modulus $q$. The generalized Bernoulli numbers $B_{k,\chi}$ attached to $\chi$ are defined as the Taylor coefficients of $\sum_{a=1}^q \chi(a) \frac{te^{at}}{e^{qt}-1} = \sum_{k=0}^\infty B_{k,\chi}\frac{t^k}{k!}$.
Parameters
$q$
—   modulus of $\chi$ ($q \geq 1$)
$n$
—   Conrey index of $\chi$ ($1 \leq n \leq \max(q-1,1)$)
$k$
—   integer ($k \geq 0$)
Comments
(1)
If $\chi(-1) \neq (-1)^k$, we have $B_{k,\chi}=0$ unless $(q,n)=(1,1)$ (that is, $\chi=1$). Thus we skip these trivial zero entries.
Programs
(P1)
Sage
chi = DirichletGroup(3).an_element()
numbers = [chi.bernoulli(n) for n in [0..10]]
Links
Data properties
Entries are of type: complex number