| L(s) = 1 | − 64·4-s + 3.59e3·9-s − 1.02e4·11-s + 4.09e3·16-s − 9.10e4·19-s − 4.63e5·29-s − 1.60e5·31-s − 2.29e5·36-s + 1.16e6·41-s + 6.58e5·44-s + 1.63e6·49-s − 4.92e5·59-s + 1.78e6·61-s − 2.62e5·64-s − 4.07e5·71-s + 5.82e6·76-s − 1.01e7·79-s + 8.10e6·81-s − 1.96e6·89-s − 3.69e7·99-s − 3.07e7·101-s + 1.40e7·109-s + 2.96e7·116-s + 4.05e7·121-s + 1.02e7·124-s + ⋯ |
| L(s) = 1 | − 1/2·4-s + 1.64·9-s − 2.33·11-s + 1/4·16-s − 3.04·19-s − 3.52·29-s − 0.966·31-s − 0.820·36-s + 2.65·41-s + 1.16·44-s + 1.98·49-s − 0.312·59-s + 1.00·61-s − 1/8·64-s − 0.135·71-s + 1.52·76-s − 2.30·79-s + 1.69·81-s − 0.294·89-s − 3.82·99-s − 2.97·101-s + 1.04·109-s + 1.76·116-s + 2.07·121-s + 0.483·124-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2500 ^{s/2} \, \Gamma_{\C}(s+7/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.6307316715\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6307316715\) |
| \(L(\frac{9}{2})\) |
|
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 | $C_2$ | \( 1 + p^{6} T^{2} \) |
| 5 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 - 3590 T^{2} + p^{14} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 1636270 T^{2} + p^{14} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 468 p T + p^{7} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 51502630 T^{2} + p^{14} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 409642270 T^{2} + p^{14} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 45500 T + p^{7} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 1615277710 T^{2} + p^{14} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 231510 T + p^{7} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 80128 T + p^{7} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 178911294550 T^{2} + p^{14} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 584922 T + p^{7} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 89233940810 T^{2} + p^{14} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 832056400030 T^{2} + p^{14} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 97027642870 T^{2} + p^{14} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 246420 T + p^{7} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 893942 T + p^{7} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 6660620719750 T^{2} + p^{14} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 203688 T + p^{7} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 7611429325390 T^{2} + p^{14} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 5053040 T + p^{7} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 54270032457190 T^{2} + p^{14} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 980010 T + p^{7} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 134058780414910 T^{2} + p^{14} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.66110684759384802754078146979, −13.25415868998954707930197309206, −13.23765420444522547317851236893, −12.71891434485489186302473515121, −12.55439816034855009633410290517, −11.02753860223303643050578208876, −10.81765355130472107910698412478, −10.32611485345816100358779920323, −9.582406321740937739307670015052, −8.977161296737043291149047793133, −8.159001958358783394679944285300, −7.50461306503744213548939534774, −7.12518294029211918821011776230, −5.86840165590916240872969206380, −5.39243756830895796066085678100, −4.20232047123596577920597714503, −4.08643021948200443550500114928, −2.46133041771981549717188026344, −1.84462955578010469944231168137, −0.30345723101526765699250942703,
0.30345723101526765699250942703, 1.84462955578010469944231168137, 2.46133041771981549717188026344, 4.08643021948200443550500114928, 4.20232047123596577920597714503, 5.39243756830895796066085678100, 5.86840165590916240872969206380, 7.12518294029211918821011776230, 7.50461306503744213548939534774, 8.159001958358783394679944285300, 8.977161296737043291149047793133, 9.582406321740937739307670015052, 10.32611485345816100358779920323, 10.81765355130472107910698412478, 11.02753860223303643050578208876, 12.55439816034855009633410290517, 12.71891434485489186302473515121, 13.23765420444522547317851236893, 13.25415868998954707930197309206, 14.66110684759384802754078146979
