| L(s) = 1 | + 186·9-s + 3.25e3·13-s + 8.46e3·25-s − 6.37e3·37-s + 6.43e4·49-s + 9.54e4·61-s + 7.03e4·73-s − 2.44e4·81-s − 1.96e5·97-s − 5.62e5·109-s + 6.05e5·117-s − 5.15e4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 5.14e6·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
| L(s) = 1 | + 0.765·9-s + 5.34·13-s + 2.70·25-s − 0.765·37-s + 3.82·49-s + 3.28·61-s + 1.54·73-s − 0.414·81-s − 2.11·97-s − 4.53·109-s + 4.09·117-s − 0.319·121-s + 5.50e−6·127-s + 5.09e−6·131-s + 4.55e−6·137-s + 4.38e−6·139-s + 3.69e−6·149-s + 3.56e−6·151-s + 3.23e−6·157-s + 2.94e−6·163-s + 2.77e−6·167-s + 13.8·169-s + 2.54e−6·173-s + 2.33e−6·179-s + 2.26e−6·181-s + 1.98e−6·191-s + 1.93e−6·193-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5308416 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5308416 ^{s/2} \, \Gamma_{\C}(s+5/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(7.191091707\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.191091707\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | $C_2^2$ | \( 1 - 62 p T^{2} + p^{10} T^{4} \) |
| good | 5 | $C_2^2$ | \( ( 1 - 4234 T^{2} + p^{10} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 - 32162 T^{2} + p^{10} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 + 25750 T^{2} + p^{10} T^{4} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 814 T + p^{5} T^{2} )^{4} \) |
| 17 | $C_2^2$ | \( ( 1 + 1063262 T^{2} + p^{10} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 - 4950746 T^{2} + p^{10} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 + 10913134 T^{2} + p^{10} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 11506042 T^{2} + p^{10} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + 51390670 T^{2} + p^{10} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 1594 T + p^{5} T^{2} )^{4} \) |
| 41 | $C_2^2$ | \( ( 1 - 152677138 T^{2} + p^{10} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - 212459498 T^{2} + p^{10} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + 186868702 T^{2} + p^{10} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 465364330 T^{2} + p^{10} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 654104714 T^{2} + p^{10} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 23870 T + p^{5} T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 - 1776663866 T^{2} + p^{10} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + 2531406670 T^{2} + p^{10} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 17578 T + p^{5} T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 - 6103337810 T^{2} + p^{10} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + 7613898598 T^{2} + p^{10} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 3344224498 T^{2} + p^{10} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 49070 T + p^{5} T^{2} )^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.78772308146863531090453958744, −10.35186458525617025511995313036, −10.30598976467433910681421463368, −9.641572493361603677967196774603, −9.194806836083580873597135303285, −8.757973595306703711585838094630, −8.754173856698912193403029870706, −8.434703029935748316592360258113, −8.267134586331697642071657745275, −7.70713941062885028853885377587, −6.99185421639082582739350543221, −6.82509346594668318358843787237, −6.65584359787269791099288835163, −6.05960487596838311017661844456, −5.86968621551603737201866094265, −5.33811882272077207077625118109, −5.04383940263279580876762042407, −3.94754323134692544541969154430, −3.89586078918740177900650871858, −3.89270305873138975659442557744, −3.07177550379763901673704327626, −2.44091914934268270490977443589, −1.41482110293227740910872288535, −1.04824361104573351254038520337, −0.935959272208809740344281422815,
0.935959272208809740344281422815, 1.04824361104573351254038520337, 1.41482110293227740910872288535, 2.44091914934268270490977443589, 3.07177550379763901673704327626, 3.89270305873138975659442557744, 3.89586078918740177900650871858, 3.94754323134692544541969154430, 5.04383940263279580876762042407, 5.33811882272077207077625118109, 5.86968621551603737201866094265, 6.05960487596838311017661844456, 6.65584359787269791099288835163, 6.82509346594668318358843787237, 6.99185421639082582739350543221, 7.70713941062885028853885377587, 8.267134586331697642071657745275, 8.434703029935748316592360258113, 8.754173856698912193403029870706, 8.757973595306703711585838094630, 9.194806836083580873597135303285, 9.641572493361603677967196774603, 10.30598976467433910681421463368, 10.35186458525617025511995313036, 10.78772308146863531090453958744
