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Values of Dedekind zeta functions of real quadratic fields at negative odd integers
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Numbers
$D$
$s$ 
$\zeta_K(s)$
5
-1:
1/30
comment: $\mathbb{Q}(\sqrt{5})$
5
-3:
1/60
comment: $\mathbb{Q}(\sqrt{5})$
5
-5:
67/630
comment: $\mathbb{Q}(\sqrt{5})$
8
-1:
1/12
comment: $\mathbb{Q}(\sqrt{2})$
8
-3:
11/120
comment: $\mathbb{Q}(\sqrt{2})$
8
-5:
361/252
comment: $\mathbb{Q}(\sqrt{2})$
12
-1:
1/6
comment: $\mathbb{Q}(\sqrt{3})$
12
-3:
23/60
comment: $\mathbb{Q}(\sqrt{3})$
12
-5:
1681/126
comment: $\mathbb{Q}(\sqrt{3})$
13
-1:
1/6
comment: $\mathbb{Q}(\sqrt{13})$
13
-3:
29/60
comment: $\mathbb{Q}(\sqrt{13})$
13
-5:
33463/1638
comment: $\mathbb{Q}(\sqrt{13})$
17
-1:
1/3
comment: $\mathbb{Q}(\sqrt{17})$
17
-3:
41/30
comment: $\mathbb{Q}(\sqrt{17})$
17
-5:
5791/63
comment: $\mathbb{Q}(\sqrt{17})$
21
-1:
1/3
comment: $\mathbb{Q}(\sqrt{21})$
21
-3:
77/30
comment: $\mathbb{Q}(\sqrt{21})$
21
-5:
17971/63
comment: $\mathbb{Q}(\sqrt{21})$
24
-1:
1/2
comment: $\mathbb{Q}(\sqrt{6})$
24
-3:
87/20
comment: $\mathbb{Q}(\sqrt{6})$
24
-5:
3623/6
comment: $\mathbb{Q}(\sqrt{6})$
28
-1:
2/3
comment: $\mathbb{Q}(\sqrt{7})$
28
-3:
113/15
comment: $\mathbb{Q}(\sqrt{7})$
28
-5:
88922/63
comment: $\mathbb{Q}(\sqrt{7})$
29
-1:
1/2
comment: $\mathbb{Q}(\sqrt{29})$
29
-3:
157/20
comment: $\mathbb{Q}(\sqrt{29})$
29
-5:
23537/14
comment: $\mathbb{Q}(\sqrt{29})$
33
-1:
1
comment: $\mathbb{Q}(\sqrt{33})$
33
-3:
141/10
comment: $\mathbb{Q}(\sqrt{33})$
33
-5:
74231/21
comment: $\mathbb{Q}(\sqrt{33})$
37
-1:
5/6
comment: $\mathbb{Q}(\sqrt{37})$
37
-3:
1129/60
comment: $\mathbb{Q}(\sqrt{37})$
37
-5:
115865/18
comment: $\mathbb{Q}(\sqrt{37})$
40
-1:
7/6
comment: $\mathbb{Q}(\sqrt{10})$
40
-3:
1577/60
comment: $\mathbb{Q}(\sqrt{10})$
40
-5:
1264807/126
comment: $\mathbb{Q}(\sqrt{10})$
41
-1:
4/3
comment: $\mathbb{Q}(\sqrt{41})$
41
-3:
448/15
comment: $\mathbb{Q}(\sqrt{41})$
41
-5:
733924/63
comment: $\mathbb{Q}(\sqrt{41})$
44
-1:
7/6
comment: $\mathbb{Q}(\sqrt{11})$
44
-3:
2153/60
comment: $\mathbb{Q}(\sqrt{11})$
44
-5:
2130727/126
comment: $\mathbb{Q}(\sqrt{11})$
53
-1:
7/6
comment: $\mathbb{Q}(\sqrt{53})$
53
-3:
775/12
comment: $\mathbb{Q}(\sqrt{53})$
53
-5:
5838037/126
comment: $\mathbb{Q}(\sqrt{53})$
56
-1:
5/3
comment: $\mathbb{Q}(\sqrt{14})$
56
-3:
2503/30
comment: $\mathbb{Q}(\sqrt{14})$
56
-5:
4013645/63
comment: $\mathbb{Q}(\sqrt{14})$
57
-1:
7/3
comment: $\mathbb{Q}(\sqrt{57})$
57
-3:
2867/30
comment: $\mathbb{Q}(\sqrt{57})$
57
-5:
4499857/63
comment: $\mathbb{Q}(\sqrt{57})$
60
-1:
2
comment: $\mathbb{Q}(\sqrt{15})$
60
-3:
537/5
comment: $\mathbb{Q}(\sqrt{15})$
60
-5:
1957882/21
comment: $\mathbb{Q}(\sqrt{15})$
61
-1:
11/6
comment: $\mathbb{Q}(\sqrt{61})$
61
-3:
6511/60
comment: $\mathbb{Q}(\sqrt{61})$
61
-5:
12685361/126
comment: $\mathbb{Q}(\sqrt{61})$
65
-1:
8/3
comment: $\mathbb{Q}(\sqrt{65})$
65
-3:
2246/15
comment: $\mathbb{Q}(\sqrt{65})$
65
-5:
9254288/63
comment: $\mathbb{Q}(\sqrt{65})$
69
-1:
2
comment: $\mathbb{Q}(\sqrt{69})$
69
-3:
165
comment: $\mathbb{Q}(\sqrt{69})$
69
-5:
4158262/21
comment: $\mathbb{Q}(\sqrt{69})$
73
-1:
11/3
comment: $\mathbb{Q}(\sqrt{73})$
73
-3:
1379/6
comment: $\mathbb{Q}(\sqrt{73})$
73
-5:
2509883/9
comment: $\mathbb{Q}(\sqrt{73})$
76
-1:
19/6
comment: $\mathbb{Q}(\sqrt{19})$
76
-3:
14933/60
comment: $\mathbb{Q}(\sqrt{19})$
76
-5:
43171459/126
comment: $\mathbb{Q}(\sqrt{19})$
77
-1:
2
comment: $\mathbb{Q}(\sqrt{77})$
77
-3:
1193/5
comment: $\mathbb{Q}(\sqrt{77})$
77
-5:
7590902/21
comment: $\mathbb{Q}(\sqrt{77})$
85
-1:
3
comment: $\mathbb{Q}(\sqrt{85})$
85
-3:
3463/10
comment: $\mathbb{Q}(\sqrt{85})$
85
-5:
13110553/21
comment: $\mathbb{Q}(\sqrt{85})$
88
-1:
23/6
comment: $\mathbb{Q}(\sqrt{22})$
88
-3:
24889/60
comment: $\mathbb{Q}(\sqrt{22})$
88
-5:
96678263/126
comment: $\mathbb{Q}(\sqrt{22})$
89
-1:
13/3
comment: $\mathbb{Q}(\sqrt{89})$
89
-3:
2701/6
comment: $\mathbb{Q}(\sqrt{89})$
89
-5:
7445389/9
comment: $\mathbb{Q}(\sqrt{89})$
92
-1:
10/3
comment: $\mathbb{Q}(\sqrt{23})$
92
-3:
7093/15
comment: $\mathbb{Q}(\sqrt{23})$
92
-5:
8794030/9
comment: $\mathbb{Q}(\sqrt{23})$
93
-1:
3
comment: $\mathbb{Q}(\sqrt{93})$
93
-3:
4679/10
comment: $\mathbb{Q}(\sqrt{93})$
93
-5:
21470873/21
comment: $\mathbb{Q}(\sqrt{93})$
97
-1:
17/3
comment: $\mathbb{Q}(\sqrt{97})$
97
-3:
18649/30
comment: $\mathbb{Q}(\sqrt{97})$
97
-5:
83892047/63
comment: $\mathbb{Q}(\sqrt{97})$
101
-1:
19/6
comment: $\mathbb{Q}(\sqrt{101})$
101
-3:
37103/60
comment: $\mathbb{Q}(\sqrt{101})$
101
-5:
28937887/18
comment: $\mathbb{Q}(\sqrt{101})$
104
-1:
25/6
comment: $\mathbb{Q}(\sqrt{26})$
104
-3:
43679/60
comment: $\mathbb{Q}(\sqrt{26})$
104
-5:
241665385/126
comment: $\mathbb{Q}(\sqrt{26})$
105
-1:
6
comment: $\mathbb{Q}(\sqrt{105})$
105
-3:
4059/5
comment: $\mathbb{Q}(\sqrt{105})$
105
-5:
14394942/7
comment: $\mathbb{Q}(\sqrt{105})$
109
-1:
9/2
comment: $\mathbb{Q}(\sqrt{109})$
109
-3:
3313/4
comment: $\mathbb{Q}(\sqrt{109})$
109
-5:
102972299/42
comment: $\mathbb{Q}(\sqrt{109})$
113
-1:
6
comment: $\mathbb{Q}(\sqrt{113})$
113
-3:
5179/5
comment: $\mathbb{Q}(\sqrt{113})$
113
-5:
64580666/21
comment: $\mathbb{Q}(\sqrt{113})$
120
-1:
17/3
comment: $\mathbb{Q}(\sqrt{30})$
120
-3:
36451/30
comment: $\mathbb{Q}(\sqrt{30})$
120
-5:
265810697/63
comment: $\mathbb{Q}(\sqrt{30})$
124
-1:
20/3
comment: $\mathbb{Q}(\sqrt{31})$
124
-3:
20714/15
comment: $\mathbb{Q}(\sqrt{31})$
124
-5:
318795140/63
comment: $\mathbb{Q}(\sqrt{31})$
129
-1:
25/3
comment: $\mathbb{Q}(\sqrt{129})$
129
-3:
50117/30
comment: $\mathbb{Q}(\sqrt{129})$
129
-5:
401956255/63
comment: $\mathbb{Q}(\sqrt{129})$
133
-1:
17/3
comment: $\mathbb{Q}(\sqrt{133})$
133
-3:
49693/30
comment: $\mathbb{Q}(\sqrt{133})$
133
-5:
461414627/63
comment: $\mathbb{Q}(\sqrt{133})$
136
-1:
23/3
comment: $\mathbb{Q}(\sqrt{34})$
136
-3:
57241/30
comment: $\mathbb{Q}(\sqrt{34})$
136
-5:
529854263/63
comment: $\mathbb{Q}(\sqrt{34})$
137
-1:
8
comment: $\mathbb{Q}(\sqrt{137})$
137
-3:
10162/5
comment: $\mathbb{Q}(\sqrt{137})$
137
-5:
62088736/7
comment: $\mathbb{Q}(\sqrt{137})$
140
-1:
19/3
comment: $\mathbb{Q}(\sqrt{35})$
140
-3:
61733/30
comment: $\mathbb{Q}(\sqrt{35})$
140
-5:
619698979/63
comment: $\mathbb{Q}(\sqrt{35})$
141
-1:
6
comment: $\mathbb{Q}(\sqrt{141})$
141
-3:
10071/5
comment: $\mathbb{Q}(\sqrt{141})$
141
-5:
70604582/7
comment: $\mathbb{Q}(\sqrt{141})$
145
-1:
32/3
comment: $\mathbb{Q}(\sqrt{145})$
145
-3:
38138/15
comment: $\mathbb{Q}(\sqrt{145})$
145
-5:
109377896/9
comment: $\mathbb{Q}(\sqrt{145})$
149
-1:
35/6
comment: $\mathbb{Q}(\sqrt{149})$
149
-3:
144799/60
comment: $\mathbb{Q}(\sqrt{149})$
149
-5:
245601455/18
comment: $\mathbb{Q}(\sqrt{149})$
152
-1:
41/6
comment: $\mathbb{Q}(\sqrt{38})$
152
-3:
32867/12
comment: $\mathbb{Q}(\sqrt{38})$
152
-5:
1948118201/126
comment: $\mathbb{Q}(\sqrt{38})$
156
-1:
26/3
comment: $\mathbb{Q}(\sqrt{39})$
156
-3:
9145/3
comment: $\mathbb{Q}(\sqrt{39})$
156
-5:
160764638/9
comment: $\mathbb{Q}(\sqrt{39})$
157
-1:
43/6
comment: $\mathbb{Q}(\sqrt{157})$
157
-3:
177551/60
comment: $\mathbb{Q}(\sqrt{157})$
157
-5:
2298167953/126
comment: $\mathbb{Q}(\sqrt{157})$
161
-1:
32/3
comment: $\mathbb{Q}(\sqrt{161})$
161
-3:
10756/3
comment: $\mathbb{Q}(\sqrt{161})$
161
-5:
1357956752/63
comment: $\mathbb{Q}(\sqrt{161})$
165
-1:
22/3
comment: $\mathbb{Q}(\sqrt{165})$
165
-3:
52291/15
comment: $\mathbb{Q}(\sqrt{165})$
165
-5:
1508356882/63
comment: $\mathbb{Q}(\sqrt{165})$
168
-1:
9
comment: $\mathbb{Q}(\sqrt{42})$
168
-3:
7875/2
comment: $\mathbb{Q}(\sqrt{42})$
168
-5:
187933043/7
comment: $\mathbb{Q}(\sqrt{42})$
172
-1:
21/2
comment: $\mathbb{Q}(\sqrt{43})$
172
-3:
86603/20
comment: $\mathbb{Q}(\sqrt{43})$
172
-5:
1285165781/42
comment: $\mathbb{Q}(\sqrt{43})$
173
-1:
13/2
comment: $\mathbb{Q}(\sqrt{173})$
173
-3:
81081/20
comment: $\mathbb{Q}(\sqrt{173})$
173
-5:
1302788663/42
comment: $\mathbb{Q}(\sqrt{173})$
177
-1:
13
comment: $\mathbb{Q}(\sqrt{177})$
177
-3:
50433/10
comment: $\mathbb{Q}(\sqrt{177})$
177
-5:
763166363/21
comment: $\mathbb{Q}(\sqrt{177})$
181
-1:
19/2
comment: $\mathbb{Q}(\sqrt{181})$
181
-3:
97679/20
comment: $\mathbb{Q}(\sqrt{181})$
181
-5:
1675329049/42
comment: $\mathbb{Q}(\sqrt{181})$
184
-1:
37/3
comment: $\mathbb{Q}(\sqrt{46})$
184
-3:
164999/30
comment: $\mathbb{Q}(\sqrt{46})$
184
-5:
2793813037/63
comment: $\mathbb{Q}(\sqrt{46})$
185
-1:
38/3
comment: $\mathbb{Q}(\sqrt{185})$
185
-3:
17461/3
comment: $\mathbb{Q}(\sqrt{185})$
185
-5:
2915797058/63
comment: $\mathbb{Q}(\sqrt{185})$
188
-1:
28/3
comment: $\mathbb{Q}(\sqrt{47})$
188
-3:
86446/15
comment: $\mathbb{Q}(\sqrt{47})$
188
-5:
3135548908/63
comment: $\mathbb{Q}(\sqrt{47})$
193
-1:
49/3
comment: $\mathbb{Q}(\sqrt{193})$
193
-3:
207353/30
comment: $\mathbb{Q}(\sqrt{193})$
193
-5:
3690186799/63
comment: $\mathbb{Q}(\sqrt{193})$
197
-1:
49/6
comment: $\mathbb{Q}(\sqrt{197})$
197
-3:
383573/60
comment: $\mathbb{Q}(\sqrt{197})$
197
-5:
7985702419/126
comment: $\mathbb{Q}(\sqrt{197})$
201
-1:
49/3
comment: $\mathbb{Q}(\sqrt{201})$
201
-3:
236669/30
comment: $\mathbb{Q}(\sqrt{201})$
201
-5:
4608016519/63
comment: $\mathbb{Q}(\sqrt{201})$
204
-1:
13
comment: $\mathbb{Q}(\sqrt{51})$
204
-3:
15591/2
comment: $\mathbb{Q}(\sqrt{51})$
204
-5:
1640393453/21
comment: $\mathbb{Q}(\sqrt{51})$
205
-1:
34/3
comment: $\mathbb{Q}(\sqrt{205})$
205
-3:
113173/15
comment: $\mathbb{Q}(\sqrt{205})$
205
-5:
4984075174/63
comment: $\mathbb{Q}(\sqrt{205})$
209
-1:
47/3
comment: $\mathbb{Q}(\sqrt{209})$
209
-3:
268003/30
comment: $\mathbb{Q}(\sqrt{209})$
209
-5:
5703545177/63
comment: $\mathbb{Q}(\sqrt{209})$
213
-1:
10
comment: $\mathbb{Q}(\sqrt{213})$
213
-3:
42501/5
comment: $\mathbb{Q}(\sqrt{213})$
213
-5:
2047735790/21
comment: $\mathbb{Q}(\sqrt{213})$
217
-1:
58/3
comment: $\mathbb{Q}(\sqrt{217})$
217
-3:
31241/3
comment: $\mathbb{Q}(\sqrt{217})$
217
-5:
7030856038/63
comment: $\mathbb{Q}(\sqrt{217})$
220
-1:
46/3
comment: $\mathbb{Q}(\sqrt{55})$
220
-3:
153847/15
comment: $\mathbb{Q}(\sqrt{55})$
220
-5:
7464304726/63
comment: $\mathbb{Q}(\sqrt{55})$
221
-1:
32/3
comment: $\mathbb{Q}(\sqrt{221})$
221
-3:
143876/15
comment: $\mathbb{Q}(\sqrt{221})$
221
-5:
7515061712/63
comment: $\mathbb{Q}(\sqrt{221})$
229
-1:
27/2
comment: $\mathbb{Q}(\sqrt{229})$
229
-3:
222503/20
comment: $\mathbb{Q}(\sqrt{229})$
229
-5:
6108899537/42
comment: $\mathbb{Q}(\sqrt{229})$
232
-1:
33/2
comment: $\mathbb{Q}(\sqrt{58})$
232
-3:
246839/20
comment: $\mathbb{Q}(\sqrt{58})$
232
-5:
6664029233/42
comment: $\mathbb{Q}(\sqrt{58})$
233
-1:
53/3
comment: $\mathbb{Q}(\sqrt{233})$
233
-3:
391129/30
comment: $\mathbb{Q}(\sqrt{233})$
233
-5:
10369299923/63
comment: $\mathbb{Q}(\sqrt{233})$
236
-1:
85/6
comment: $\mathbb{Q}(\sqrt{59})$
236
-3:
768827/60
comment: $\mathbb{Q}(\sqrt{59})$
236
-5:
21905188405/126
comment: $\mathbb{Q}(\sqrt{59})$
237
-1:
35/3
comment: $\mathbb{Q}(\sqrt{237})$
237
-3:
370519/30
comment: $\mathbb{Q}(\sqrt{237})$
237
-5:
11051543705/63
comment: $\mathbb{Q}(\sqrt{237})$
241
-1:
71/3
comment: $\mathbb{Q}(\sqrt{241})$
241
-3:
90443/6
comment: $\mathbb{Q}(\sqrt{241})$
241
-5:
12520598441/63
comment: $\mathbb{Q}(\sqrt{241})$
248
-1:
14
comment: $\mathbb{Q}(\sqrt{62})$
248
-3:
15193
comment: $\mathbb{Q}(\sqrt{62})$
248
-5:
4795214654/21
comment: $\mathbb{Q}(\sqrt{62})$
249
-1:
23
comment: $\mathbb{Q}(\sqrt{249})$
249
-3:
167043/10
comment: $\mathbb{Q}(\sqrt{249})$
249
-5:
4987900033/21
comment: $\mathbb{Q}(\sqrt{249})$
253
-1:
15
comment: $\mathbb{Q}(\sqrt{253})$
253
-3:
157307/10
comment: $\mathbb{Q}(\sqrt{253})$
253
-5:
754838795/3
comment: $\mathbb{Q}(\sqrt{253})$
257
-1:
20
comment: $\mathbb{Q}(\sqrt{257})$
257
-3:
18362
comment: $\mathbb{Q}(\sqrt{257})$
257
-5:
1975489220/7
comment: $\mathbb{Q}(\sqrt{257})$
264
-1:
56/3
comment: $\mathbb{Q}(\sqrt{66})$
264
-3:
288086/15
comment: $\mathbb{Q}(\sqrt{66})$
264
-5:
20319726416/63
comment: $\mathbb{Q}(\sqrt{66})$
265
-1:
80/3
comment: $\mathbb{Q}(\sqrt{265})$
265
-3:
314762/15
comment: $\mathbb{Q}(\sqrt{265})$
265
-5:
21103590920/63
comment: $\mathbb{Q}(\sqrt{265})$
268
-1:
41/2
comment: $\mathbb{Q}(\sqrt{67})$
268
-3:
408959/20
comment: $\mathbb{Q}(\sqrt{67})$
268
-5:
4911035827/14
comment: $\mathbb{Q}(\sqrt{67})$
269
-1:
83/6
comment: $\mathbb{Q}(\sqrt{269})$
269
-3:
1144087/60
comment: $\mathbb{Q}(\sqrt{269})$
269
-5:
44303218313/126
comment: $\mathbb{Q}(\sqrt{269})$
273
-1:
74/3
comment: $\mathbb{Q}(\sqrt{273})$
273
-3:
344597/15
comment: $\mathbb{Q}(\sqrt{273})$
273
-5:
24818669654/63
comment: $\mathbb{Q}(\sqrt{273})$
277
-1:
103/6
comment: $\mathbb{Q}(\sqrt{277})$
277
-3:
1296131/60
comment: $\mathbb{Q}(\sqrt{277})$
277
-5:
52189035253/126
comment: $\mathbb{Q}(\sqrt{277})$
280
-1:
67/3
comment: $\mathbb{Q}(\sqrt{70})$
280
-3:
143185/6
comment: $\mathbb{Q}(\sqrt{70})$
280
-5:
28121438227/63
comment: $\mathbb{Q}(\sqrt{70})$
281
-1:
25
comment: $\mathbb{Q}(\sqrt{281})$
281
-3:
251933/10
comment: $\mathbb{Q}(\sqrt{281})$
281
-5:
9685249135/21
comment: $\mathbb{Q}(\sqrt{281})$
284
-1:
58/3
comment: $\mathbb{Q}(\sqrt{71})$
284
-3:
367741/15
comment: $\mathbb{Q}(\sqrt{71})$
284
-5:
30322094578/63
comment: $\mathbb{Q}(\sqrt{71})$
285
-1:
16
comment: $\mathbb{Q}(\sqrt{285})$
285
-3:
117948/5
comment: $\mathbb{Q}(\sqrt{285})$
285
-5:
10159218656/21
comment: $\mathbb{Q}(\sqrt{285})$
293
-1:
85/6
comment: $\mathbb{Q}(\sqrt{293})$
293
-3:
1537769/60
comment: $\mathbb{Q}(\sqrt{293})$
293
-5:
70878320575/126
comment: $\mathbb{Q}(\sqrt{293})$
296
-1:
41/2
comment: $\mathbb{Q}(\sqrt{74})$
296
-3:
566711/20
comment: $\mathbb{Q}(\sqrt{74})$
296
-5:
1208653701/2
comment: $\mathbb{Q}(\sqrt{74})$
301
-1:
62/3
comment: $\mathbb{Q}(\sqrt{301})$
301
-3:
434639/15
comment: $\mathbb{Q}(\sqrt{301})$
301
-5:
41217199082/63
comment: $\mathbb{Q}(\sqrt{301})$
305
-1:
82/3
comment: $\mathbb{Q}(\sqrt{305})$
305
-3:
502651/15
comment: $\mathbb{Q}(\sqrt{305})$
305
-5:
45600398182/63
comment: $\mathbb{Q}(\sqrt{305})$
309
-1:
59/3
comment: $\mathbb{Q}(\sqrt{309})$
309
-3:
941551/30
comment: $\mathbb{Q}(\sqrt{309})$
309
-5:
47549078369/63
comment: $\mathbb{Q}(\sqrt{309})$
312
-1:
23
comment: $\mathbb{Q}(\sqrt{78})$
312
-3:
343821/10
comment: $\mathbb{Q}(\sqrt{78})$
312
-5:
16973725423/21
comment: $\mathbb{Q}(\sqrt{78})$
313
-1:
100/3
comment: $\mathbb{Q}(\sqrt{313})$
313
-3:
562816/15
comment: $\mathbb{Q}(\sqrt{313})$
313
-5:
52719381700/63
comment: $\mathbb{Q}(\sqrt{313})$
316
-1:
28
comment: $\mathbb{Q}(\sqrt{79})$
316
-3:
182558/5
comment: $\mathbb{Q}(\sqrt{79})$
316
-5:
6077653476/7
comment: $\mathbb{Q}(\sqrt{79})$
317
-1:
101/6
comment: $\mathbb{Q}(\sqrt{317})$
317
-3:
2027569/60
comment: $\mathbb{Q}(\sqrt{317})$
317
-5:
109288708031/126
comment: $\mathbb{Q}(\sqrt{317})$
321
-1:
33
comment: $\mathbb{Q}(\sqrt{321})$
321
-3:
406053/10
comment: $\mathbb{Q}(\sqrt{321})$
321
-5:
6721568141/7
comment: $\mathbb{Q}(\sqrt{321})$
328
-1:
27
comment: $\mathbb{Q}(\sqrt{82})$
328
-3:
82877/2
comment: $\mathbb{Q}(\sqrt{82})$
328
-5:
22378033507/21
comment: $\mathbb{Q}(\sqrt{82})$
329
-1:
94/3
comment: $\mathbb{Q}(\sqrt{329})$
329
-3:
656047/15
comment: $\mathbb{Q}(\sqrt{329})$
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-5:
8308420210459/21
comment: $\mathbb{Q}(\sqrt{965})$
969
-1:
174
comment: $\mathbb{Q}(\sqrt{969})$
969
-3:
9703827/5
comment: $\mathbb{Q}(\sqrt{969})$
969
-5:
2927345556118/7
comment: $\mathbb{Q}(\sqrt{969})$
973
-1:
335/3
comment: $\mathbb{Q}(\sqrt{973})$
973
-3:
52624747/30
comment: $\mathbb{Q}(\sqrt{973})$
973
-5:
26153379384365/63
comment: $\mathbb{Q}(\sqrt{973})$
977
-1:
457/3
comment: $\mathbb{Q}(\sqrt{977})$
977
-3:
59052041/30
comment: $\mathbb{Q}(\sqrt{977})$
977
-5:
27524047139047/63
comment: $\mathbb{Q}(\sqrt{977})$
984
-1:
421/3
comment: $\mathbb{Q}(\sqrt{246})$
984
-3:
57651131/30
comment: $\mathbb{Q}(\sqrt{246})$
984
-5:
28221293422501/63
comment: $\mathbb{Q}(\sqrt{246})$
985
-1:
192
comment: $\mathbb{Q}(\sqrt{985})$
985
-3:
10387538/5
comment: $\mathbb{Q}(\sqrt{985})$
985
-5:
9622364507992/21
comment: $\mathbb{Q}(\sqrt{985})$
988
-1:
146
comment: $\mathbb{Q}(\sqrt{247})$
988
-3:
9835897/5
comment: $\mathbb{Q}(\sqrt{247})$
988
-5:
3210443503502/7
comment: $\mathbb{Q}(\sqrt{247})$
989
-1:
301/3
comment: $\mathbb{Q}(\sqrt{989})$
989
-3:
54549569/30
comment: $\mathbb{Q}(\sqrt{989})$
989
-5:
4076206834753/9
comment: $\mathbb{Q}(\sqrt{989})$
993
-1:
167
comment: $\mathbb{Q}(\sqrt{993})$
993
-3:
21075203/10
comment: $\mathbb{Q}(\sqrt{993})$
993
-5:
3348563100699/7
comment: $\mathbb{Q}(\sqrt{993})$
997
-1:
225/2
comment: $\mathbb{Q}(\sqrt{997})$
997
-3:
38187317/20
comment: $\mathbb{Q}(\sqrt{997})$
997
-5:
19935873593795/42
comment: $\mathbb{Q}(\sqrt{997})$
Definition
Let $K=\mathbb{Q}(\sqrt{D})$ be the real quadratic field of fundamental discriminant $D>1$ and $\zeta_K(s)$ its Dedekind zeta function [6]. Listed is $\zeta_K(s)$ at negative odd integers $s$, where it is a nonzero rational number.
Parameters
$D$
—   fundamental discriminant of $K$ ($D$ a fundamental discriminant, $D>1$)
$s$
—   argument ($s$ a negative odd integer)
Formulas
(1)
$\zeta_K(1-2m)=\zeta(1-2m)\,L(1-2m,\chi_D)=\frac{B_{2m}}{2m}\cdot\frac{B_{2m,\chi_D}}{2m}$, since $\zeta(1-2m)=-B_{2m}/2m$ and $L(1-2m,\chi)=-B_{2m,\chi}/2m$. $B_{2m}$ is in the Bernoulli numbers and $B_{2m,\chi_D}$, for $D=5$ and $D=8$, in the generalized Bernoulli numbers.
(2)
$\zeta_K(-1)=\frac{1}{60}\sum_{b^2<D,\ b\equiv D \pmod 2}\sigma_1\!\left(\frac{D-b^2}{4}\right)$ ([1]; [3], where it is the case $m=1$ of Siegel's formula for $\zeta_K(1-2m)$).
(3)
$\zeta_K(2m)=\frac{2^{4m}m^2\,\pi^{4m}}{((2m)!)^2\,D^{2m-1/2}}\,\zeta_K(1-2m)$, from $\Lambda_K(s)=D^{s/2}\bigl(\pi^{-s/2}\Gamma(s/2)\bigr)^2\zeta_K(s)=\Lambda_K(1-s)$; in particular $\zeta_K(2)=4\pi^4\zeta_K(-1)/D^{3/2}$, so $\zeta_{\mathbb{Q}(\sqrt{5})}(2)=2\pi^4/(75\sqrt{5})$. The rationals $\zeta_K(2m)\sqrt{D}/\pi^{4m}$ are [7] and [8].
(4)
$2\zeta_K(-1)$ is the Euler number, in the orbifold sense of Gauss–Bonnet, of the Hilbert modular surface $\mathrm{SL}_2(\mathcal{O}_K)\backslash\mathfrak{H}^2$ ([4], [5]); for $K=\mathbb{Q}(\sqrt{5})$ it is $1/15$. That is the two-dimensional analogue of $\zeta(-1)=-1/12$ being the Euler characteristic of $\mathrm{SL}_2(\mathbb{Z})$.
Comments
(5)
That $\zeta_K(1-2m)$ is rational for every totally real field $K$ and every $m\geq 1$ is the Siegel–Klingen theorem ([2], [1]); for a real quadratic field it follows from the factorisation $\zeta_K(s)=\zeta(s)\,L(s,\chi_D)$ and the Bernoulli-number formula (1). Every value here is positive: $\zeta_K(2m)>0$ and the functional equation relates the two by a positive factor.
(6)
Only real quadratic fields and only negative odd $s$, because that is where the values are not zero. From the functional equation, $\zeta_K$ vanishes at every negative even integer to order $r_1+r_2$ and at every negative odd integer to order $r_2$, the number of complex places, so for an imaginary quadratic field $\zeta_K(1-2m)=0$ and for a real quadratic field it is not. At $s=0$ the order is $r_1+r_2-1=1$, so $\zeta_K(0)=0$ too.
(7)
$\zeta_K(s)=\sum_{\mathfrak{a}}N(\mathfrak{a})^{-s}$, summed over the nonzero ideals of $\mathcal{O}_K$ and continued to the whole plane. $\chi_D=\left(\frac{D}{\cdot}\right)$ is the Kronecker symbol, the primitive even quadratic character of conductor $D$; $B_n$ are the Bernoulli numbers with $B_1=-1/2$, $B_n(x)$ the Bernoulli polynomials, and $B_{n,\chi}=q^{n-1}\sum_{a=1}^{q}\chi(a)B_n(a/q)$ the generalized Bernoulli numbers of a character $\chi$ of modulus $q$; $\sigma_1(n)$ is the sum of the divisors of $n$.
(8)
$D$ is the discriminant of $K$, not the squarefree integer $d$ with $K=\mathbb{Q}(\sqrt{d})$: $D=d$ if $d\equiv 1 \pmod 4$ and $D=4d$ otherwise, so $\mathbb{Q}(\sqrt{3})$ is $D=12$. The comment on each entry names the field. Every real fundamental discriminant with $D\leq 1000$ is listed, 302 of them, the same enumeration as the residues at $s=1$.
(9)
$60\,\zeta_K(-1)$ is always an integer, by Siegel's formula (2), and $\zeta_K(-1)$ itself is an integer for 88 of the 302 fields here ($\zeta_K(-1)=1$ for $D=33$, $2$ for $D=60,69,77$). Every denominator at $s=-1$ divides $60$, at $s=-3$ divides $120$, and at $s=-5$ divides $16380$.
Programs
(P1)
Sage
D, m = 5, 1
bernoulli(2*m) * kronecker_character(D).bernoulli(2*m) / (2*m)^2     # 1/30
(P2)
PARI/GP
bestappr(lfun(lfuncreate(x^2 - 5), -1), 10^6)    \\ 1/30, numerically; the denominator divides 60
References
[1]
Carl Ludwig Siegel, "Berechnung von Zetafunktionen an ganzzahligen Stellen", Nachrichten der Akademie der Wissenschaften in Göttingen, Math.-Phys. Klasse 1969, 87–102.
[2]
Helmut Klingen, "Über die Werte der Dedekindschen Zetafunktion", Mathematische Annalen 145 (1962), 265–272.
[3]
Don Zagier, "On the values at negative integers of the zeta-function of a real quadratic field", L'Enseignement Mathématique 22 (1976), 55–95.
[4]
Friedrich Hirzebruch, "Hilbert modular surfaces", L'Enseignement Mathématique 19 (1973), 183–281.
[5]
Gerard van der Geer, "Hilbert Modular Surfaces", Ergebnisse der Mathematik und ihrer Grenzgebiete 16, Springer, 1988, Chapter IV.
Links
Similar tables
Bernoulli numbers —   the factor $B_{2m}$, and $\zeta(1-2m)=-B_{2m}/2m$
Generalized Bernoulli numbers —   the factor $B_{2m,\chi_D}$, indexed there by the Conrey label $(q,n)$ of $\chi_D$; $\chi_5$ is $(5,4)$ and $\chi_8$ is $(8,5)$
Residues of Dedekind zeta functions of quadratic fields —   the same $\zeta_K$ at $s=1$, for the same fields and the imaginary ones; its comments carry the class numbers
Values of the Riemann zeta function at rational numbers —   the factor $\zeta(1-2m)$ of $\zeta_K(1-2m)=\zeta(1-2m)L(1-2m,\chi_D)$
Bernoulli polynomials —   $B_{2m,\chi_D}$ is a sum of their values at $a/D$
$j$-invariants of elliptic curves over quadratic fields with everywhere good reduction —   indexed by the same discriminant $D$
Data properties
Entries are of type: rational number
How well the digits are known: exact
Table is complete: no (every real fundamental discriminant with $D\leq 1000$ is here, at $s=-1,-3,-5$)
How they were obtained:

Each value is $B_{2m}B_{2m,\chi_D}/(2m)^2$ in exact rational arithmetic. The generator refuses a value unless it equals Siegel's $\sigma_1$ formula at $s=-1$ and satisfies the functional equation in ball arithmetic at every $s$, with $\zeta_K(2m)$ computed independently from the Hurwitz zeta function at 256 bits (worst radius $3\cdot 10^{-74}$).

more

All 906 were also compared with PARI's lfun for 121 of the fields, with the generalized Bernoulli numbers stored in T49 for $D=5,8$, and with the OEIS array A370412/A370411.



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