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Critical values of Rankin-Selberg L-functions of level-one cusp forms

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Numbers
$k$
$i$
$\ell$, $j$
$s$ 
$L(f_{k,i}\otimes f_{\ell,j},s)$
16
1
12, 1
12:
1.764790264111159589525680806569476356180861793197522430986097214994965967507062908824912966606643003e-1
16
1
12, 1
13:
4.466097886582346732956292974976054891286031294102911115704005508452412740053850379238900986149227348e-1
16
1
12, 1
14:
6.781326054939848077418159788406417165230996865699562287825546140909601681084947210022002242584395294e-1
16
1
12, 1
15:
8.286448297159346761607758126580157677554507138161197632807947135848968239032800023794648586848325008e-1
18
1
12, 1
12:
7.120697594231555484207254022999195859767158673424945088748248180652066674524985036866005365593045565e-1
18
1
12, 1
13:
1.229873195509947938637908761219050651594710884529156651831332637037445792056212409889981875026131054e0
18
1
12, 1
14:
1.333885923427122569731199244049956739738853850530981316905585406830473995681079351983491505825829196e0
18
1
12, 1
15:
1.253802512419759249598637213389765464896878455477575438696333181720432367011547238072675498023442762e0
18
1
12, 1
16:
1.154956580836328770266686430891076847269347296859164456138047101651305542323244262505556148914217776e0
18
1
12, 1
17:
1.085615671741842668246128270197130834749771485839619648299008996094208211482315768345028918343433874e0
20
1
12, 1
12:
5.380003562880316956827779273601915299106576448831298289712299199202412886772263471256873014161606863e0
20
1
12, 1
13:
5.618889612918519170485291332591258584374476213460572668631612709574554895161562268416393793779090545e0
20
1
12, 1
14:
3.513063561721912183912475754842380556933325828381652994569066236496695793000756616539868785192561381e0
20
1
12, 1
15:
1.981288433718699550641790846063984466229282347782770946024944338345867899960175535520877452493749042e0
20
1
12, 1
16:
1.303635536350501271930719751168943802808263223316969349523750633623472478482748851283507641498447439e0
20
1
12, 1
17:
1.072197252248449491916280925225401361266740308077370577030117746699718371280375147927565440074277510e0
20
1
12, 1
18:
1.007825020916877553944971232332446159988141137535884052276641904653022423812070971514750994588520274e0
20
1
12, 1
19:
9.946834261969189613489377166947575275311582740153590998192822532528941633994618234735995757253562767e-1
22
1
12, 1
12:
9.514822365567510798862666025154094496687222456903499368310691334728797574995449222919458908763963518e1
22
1
12, 1
13:
6.886552397617828234061592982285970412536675856470898239808064661358245150185579906270509217456514013e1
22
1
12, 1
14:
2.751722585095602925738414486728140101405908415695605915982381524664308452982753807952706057108759813e1
22
1
12, 1
15:
8.621718519503666594043222222131163824264183675098653517198887290869073533092617721986565921625133083e0
22
1
12, 1
16:
2.836431701501567233702942220951151587956354955793896750943116529086101719170321079590364654137402504e0
22
1
12, 1
17:
1.399722940226462294870824968502621677716951301087918313037068646949277788077808984341149208116730458e0
22
1
12, 1
18:
1.083506799307045451828019083080164184158834953197935289208294479301726498411713199287952556113998202e0
22
1
12, 1
19:
1.018455569052481925024067452289986191036726965662364337147151221881102821117064431226954048217412954e0
22
1
12, 1
20:
1.005175356663434279018531351848027565169539028300129555196734644181446958477640907634321201461508611e0
22
1
12, 1
21:
1.002089204443636032074087914898548566318833114296618256040026576086354788535879469379789970046459500e0
24
1
12, 1
12:
3.135288133159181321527794216251230216971650153878468169743777111919566235887424421011590484779725304e3
24
1
12, 1
13:
1.706055861945800013451486248212643626324734401376158261945490931894645908876286723405596048178744566e3
24
1
12, 1
14:
5.019983601909643452535138305893083929380638286923897884845358400932580619790259611703399029715115116e2
24
1
12, 1
15:
1.110300767323345885118817630494399308689463602853271970474854498161192411243853821291151474566158514e2
24
1
12, 1
16:
2.259091508555605454695439630870223371273527057274952775124383097913730773456464838770022161584781089e1
24
1
12, 1
17:
5.441980408141968265333217181287514977282459274969012229825054760681347384502461945268490443093255150e0
24
1
12, 1
18:
2.106282109287844473620822820706310682645756514428003425600654264147103440074840456585765367809360326e0
24
1
12, 1
19:
1.351924160362957667130643346006030017339541331251871098485340372265620668560604436565737668655374644e0
24
1
12, 1
20:
1.135491222293512683054016826431793016173672067468116543312226297594148977833413021684100767975008029e0
24
1
12, 1
21:
1.058383639268302896656192413357504921487941075906800896112883151502856407071463801809433200015030566e0
24
1
12, 1
22:
1.026740982647085063076392564193525996653977407847225976724299804388896923195519618127327412980788141e0
24
1
12, 1
23:
1.012670060130833295710994869929643236897203328428848283565390049932500375901598148270208506434992457e0
24
2
12, 1
12:
3.050125073221586177915046098756800410524651228538680002881627442839483028150874636372290601979346119e3
24
2
12, 1
13:
1.611732791520152264610078031588459200794659770376074728774561986042593522192254432354256571495080399e3
24
2
12, 1
14:
4.453238474366547738408984723306210223580531635345644265746929326663611373366398141094905940853373314e2
24
2
12, 1
15:
8.563612328412571976038310458104929833540619737456645279079601030087588522296475953576125300868725284e1
24
2
12, 1
16:
1.269289782118402903660042707186894863250135136903121233059946669700857185653566354234713102965297966e1
24
2
12, 1
17:
1.725242522991605968162614535679889841413952062952600641661258867590577929918061598551080836160419563e0
24
2
12, 1
18:
6.677435763868416956252811293320978222896607493945082417094259195426152430352921468388971278050328798e-1
24
2
12, 1
19:
7.595900902858295932327562312396947830009805944923671253143480009229557697194849077136049085769675027e-1
24
2
12, 1
20:
8.757903188231476333113173962679559525398408200758264654335588463450922691829915470059250375044766219e-1
24
2
12, 1
21:
9.388944500210594628862080936751907660293769946745148380552732758504220171549986739993308121005297656e-1
24
2
12, 1
22:
9.699753372920348573275508246408794646074673987570240259990055606554880642774451878479107678149172486e-1
24
2
12, 1
23:
9.851631525149674054556502460964153150726682232506572165528677010222856655746432255527149574784257204e-1
26
1
12, 1
12:
1.710738400971672751756376412156252382685672019559611355251403218930710819281506485024996772131440831e5
26
1
12, 1
13:
7.122746687837443200149723448173327466174916454848962064365837922635677235371723068512955259116873158e4
26
1
12, 1
14:
1.548137055328998835052628000346343660791830536057743399979023702485939431346603116878435369172788077e4
26
1
12, 1
15:
2.381220825158120539385452315216337495745844387997416005735242168696860348123851871621092445586846834e3
26
1
12, 1
16:
2.984343814088408965140725160104611565025553423985336824832539060798437786951055617911130772791250726e2
26
1
12, 1
17:
3.436334164887255772792744613830928755607017386090814273900724786352209325833787230896596712968718901e1
26
1
12, 1
18:
4.750036246132350884985438030303838438020294741628100175005027861553833509412581723956665166653972518e0
26
1
12, 1
19:
1.488285036195613530199894641950979553449230928807927133533087477657695245527690495484098532205204915e0
26
1
12, 1
20:
1.082331492665069882407974202845099319580109967884735064657430259845992714690587742711134361327955018e0
26
1
12, 1
21:
1.017350825089979864542087461044893272134815559283918102479517776804507424991804597858502790438196652e0
26
1
12, 1
22:
1.004085018075995555668614068885306747805769273713237824319818552858291151287728174795189629606780639e0
26
1
12, 1
23:
1.001002213481892317079906040844902109015958292237788728236746439942755710428877459128431390182861477e0
26
1
12, 1
24:
1.000253719195842083238021155753367303609373264961863470387915887068892053231682750429139684311386309e0
26
1
12, 1
25:
1.000068879367134261580444229068647710552493108208848371810270472327518135506419154628980388456417229e0
Definition
For normalized level-one cuspidal Hecke eigenforms $f_{k,i}$ and $f_{\ell,j}$ of even weights $k>\ell$, this table gives the uncompleted arithmetic Rankin–Selberg value $L(f_{k,i}\otimes f_{\ell,j},s)$ at its critical integers [1].
Parameters
$k$
—   larger weight ($k>\ell$ is even and $\dim S_k(\mathrm{SL}_2(\mathbb Z))>0$)
$i$
—   embedded eigenform of weight $k$ ($1\leq i\leq\dim S_k(\mathrm{SL}_2(\mathbb Z))$; forms are ordered lexicographically by their real Fourier coefficients $(a_2,a_3,\ldots)$)
$\ell$
—   smaller weight ($\ell<k$ is even and $\dim S_\ell(\mathrm{SL}_2(\mathbb Z))>0$)
$j$
—   embedded eigenform of weight $\ell$ ($1\leq j\leq\dim S_\ell(\mathrm{SL}_2(\mathbb Z))$; forms use the same lexicographic Fourier-coefficient ordering)
$s$
—   critical integer ($\ell\leq s\leq k-1$)
Formulas
(1)
For a prime $p$, let $\alpha_p,\beta_p$ be the roots of $X^2-a_p(f)X+p^{k-1}$ and $\gamma_p,\delta_p$ the roots of $X^2-a_p(g)X+p^{\ell-1}$, where $f=f_{k,i}$ and $g=f_{\ell,j}$. For $\operatorname{Re}s>(k+\ell)/2$, $L(f\otimes g,s)=\prod_p\prod_{u\in\{\alpha_p,\beta_p\}}\prod_{v\in\{\gamma_p,\delta_p\}}(1-uvp^{-s})^{-1}$ [1], Proposition 4.1.2. Analytic continuation defines its other values.
(2)
Put $\Gamma_{\mathbb C}(u)=2(2\pi)^{-u}\Gamma(u)$. Then $\Lambda_\otimes(f,g,s)=\Gamma_{\mathbb C}(s)\Gamma_{\mathbb C}(s-\ell+1)L(f\otimes g,s)$ satisfies $\Lambda_\otimes(f,g,s)=\Lambda_\otimes(f,g,k+\ell-1-s)$, with conductor and root number both $1$.
(3)
For $\operatorname{Re}s>(k+\ell)/2$, $L(f\otimes g,s)=\zeta(2s-k-\ell+2)\sum_{n\geq1}a_n(f)a_n(g)n^{-s}$. Equivalently, its Dirichlet coefficient is $c_n=\sum_{d^2\mid n}d^{k+\ell-2}a_{n/d^2}(f)a_{n/d^2}(g)$. Here $\zeta$ is the Riemann zeta function.
(4)
For $k-\ell\geq4$, let $E_{k-\ell}$ be the constant-one Eisenstein series and set $\langle f,h\rangle=\int_{\mathrm{SL}_2(\mathbb Z)\backslash\mathbb H}f(z)\overline{h(z)}y^k\,dx\,dy/y^2$. Rankin unfolding gives $L(f\otimes g,k-1)=\zeta(k-\ell)(4\pi)^{k-1}\langle f,gE_{k-\ell}\rangle/(k-2)!$. The integral convention agrees with the Petersson self-products.
(5)
For even $h\geq4$, $E_h=1-\dfrac{2h}{B_h}\sum_{n\geq1}\sigma_{h-1}(n)q^n$, where $B_h$ is the Bernoulli number with $B_2=1/6$ and $\sigma_r(n)=\sum_{d\mid n}d^r$.
Comments
(6)
Write $q=e^{2\pi iz}$ and $f_{k,i}(z)=\sum_{n\geq1}a_n(f_{k,i})q^n$, with $a_1=1$. Every finite Euler factor is included. Level one has trivial character and no ramified finite factors. The arithmetic normalization has centre $(k+\ell-1)/2$.
(7)
The indices $i$ and $j$ count all real embeddings of all normalized eigenforms in their respective weights. In the selected weights the $a_2$ values are distinct, so lexicographic Fourier-coefficient ordering agrees with the increasing-$a_2$ convention of the ordinary critical-value table, the normalized Hecke-eigenvalue table and the Petersson table. The roots of the Hecke polynomial $\chi_{k,2}$ identify the weight-$k$ forms.
(8)
An integer is critical when the gamma factors in (2) are regular both at $s$ and at $k+\ell-1-s$. This gives exactly $\ell\leq s\leq k-1$. Equal-weight pairs have no critical integers in this sense and are outside the table's domain.
(9)
Write $\Delta$ for the modular discriminant, and $E_4$ and $E_6$ for the constant-one Eisenstein series of weights four and six. The selected higher-weight forms are $\Delta E_4$, $\Delta E_6$, $\Delta E_4^2$, $\Delta E_4E_6$, $\Delta E_4^3+(a_2-696)\Delta^2$, and $\Delta E_4^2E_6$ at weights $16,18,20,22,24,26$, respectively. Each is paired with $f_{12,1}=\Delta$. In weight $24$, $a_2$ runs through both roots of $X^2-1080X-20468736$, isolated by $(-4017,-4016)$ for $i=1$ and $(5096,5097)$ for $i=2$.
(10)
The convention $\Gamma_{\mathbb C}(u)=2(2\pi)^{-u}\Gamma(u)$ in (2) includes a factor $2$ absent from [1], Proposition 4.1.5. Our completion is therefore four times theirs. The uncompleted $L$-values are identical.
Programs
(P1)
PARI/GP
default(realbitprecision, 750);
k=16; s=15; h=(k+10)/2;
f=mfeigenbasis(mfinit([1,k],0))[1]; g=mfDelta();
RS(n)={my(a=mfcoefs(f,n),b=mfcoefs(g,n),c=vector(n)); for(d=1,sqrtint(n),for(m=1,n\d^2,c[d^2*m]+=d^(k+10)*a[m+1]*b[m+1])); vector(n,j,c[j]/j^h)};
L=lfuncreate([RS,0,[h,h+1,h-11,h-10],1,1,1]);
lfun(L,s-h)  \\ L(f_(16,1) tensor Delta,15) = 0.828644829715934676...
References
[1]
A. Lei, D. Loeffler and S. L. Zerbes, Euler systems for Rankin–Selberg convolutions of modular forms, Annals of Mathematics 180 (2014), 653–771, sections 4.1 and 4.2. (doi)
Links
Similar tables
Critical values of symmetric-square L-functions —   degree-three symmetric-square values for the same eigenform roster; this table pairs each higher-weight form with Delta
Fourier coefficients of the modular discriminant —   the coefficients of the fixed weight-twelve factor Delta
Hecke polynomials of level-one cusp forms —   the roots of the weight-k Hecke polynomial at prime 2 identify and order the embedded eigenforms
Normalized Hecke eigenvalues —   stores $a_p/p^{(k-1)/2}$; multiplying by $p^{(k-1)/2}$ recovers the arithmetic coefficients in (1)
Ordinary critical L-values —   degree-two values of the individual factors, with the same Fourier normalization and embedding order
Petersson norms of level-one cusp forms —   the self-products enter the endpoint relation (4) when $gE_{k-\ell}=f$
Critical L-values of higher-level newforms —   ordinary critical values with additional levels, characters and coefficient fields
Petersson norms of higher-level newforms —   Petersson self-products beyond level one, where tensor constructions require additional ramified-factor conventions
Data properties
Entries are of type: real number
How well the digits are known: heuristic (agreement-checked)
Table is complete: no (it holds all $66$ critical values for $f_{k,i}\otimes\Delta$ in every nonempty even-weight cusp space with $12<k\leq26$, including both embeddings in weight $24$. The anchor $\Delta=f_{12,1}$ is the unique minimal-weight cusp form and defines a reference slice of tensor products. The noncanonical weight bound includes all intervening spaces, the first quadratic coefficient field and the next one-dimensional space, balancing weight and coefficient-field coverage before extending to all pairs. Every integer $12\leq s\leq k-1$ is included for each selected form)
How they were obtained:

Values are recorded to $100$ significant digits after fresh PARI/GP calculations at $160$ and $220$ requested working decimal digits, using explicit bit precisions and exact coefficient data [2]. The maximum relative difference between these calculations is less than $1.3\times10^{-163}$. These are agreement checks, not certified analytic error bounds.

more

The generator builds exact Fourier coefficients from the divisor-sum expansions of $E_4$ and $E_6$ and $\Delta$, using $\Delta=(E_4^3-E_6^2)/1728$, then forms reciprocal quartic Euler factors and uses multiplicativity. It evaluates in analytic coordinates with $h=(k+\ell-2)/2$, coefficients $c_n/n^h$ and argument $s-h$, preserving the arithmetic values stored here. The quadratic embedding is refined inside each coefficient callback at the precision PARI requests. An initial unshifted weight-sixteen experiment retained only about $105$ agreeing digits; the exact coordinate shift removed that measured loss.

An independent computation constructs Sage modular-symbol eigenforms, forms the square-divisor coefficients in (3), and evaluates every critical argument using Dokchitser's L-function calculator at $700$ bits [4]. Every Fourier coefficient used by that computation agrees exactly with the generator, beyond each Sturm bound. The independent computation checks all reflected values and uses the nonzero control $\zeta(2)=\pi^2/6$ before comparing results. Its cancellation warnings are retained in the attached evidence.

Every right-endpoint value is additionally checked by (4), using PARI's independent Haberland period-pairing algorithm for $\langle f,\Delta E_{k-12}\rangle$ [3]. This includes both weight-twenty-four embeddings, without replacing the product by a self-product. Known initial coefficients of $\Delta$ control the modular-form construction. The exact dimension roster includes every embedding through weight $26$.

The attached generator, exact roster, argument-selection rationale, numerical evidence and checks record the software versions, source hashes and supported precision.

Every independent L-value and endpoint pairing agrees to at least $130$ relative decimal digits. The attached evidence records each measured difference. Dokchitser reports a maximum loss of 6 decimal digits from cancellation.



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