| L(s) = 1 | − 3.19e3·9-s + 2.16e3·11-s + 6.69e4·19-s − 2.50e5·29-s + 1.47e5·31-s − 4.53e4·41-s − 2.18e5·49-s + 2.19e6·59-s − 8.45e5·61-s + 4.57e6·71-s − 4.03e6·79-s + 5.42e6·81-s − 4.37e6·89-s − 6.92e6·99-s + 8.66e5·101-s − 3.20e7·109-s − 3.54e7·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 9.55e7·169-s + ⋯ |
| L(s) = 1 | − 1.46·9-s + 0.490·11-s + 2.23·19-s − 1.90·29-s + 0.889·31-s − 0.102·41-s − 0.265·49-s + 1.39·59-s − 0.477·61-s + 1.51·71-s − 0.921·79-s + 1.13·81-s − 0.657·89-s − 0.716·99-s + 0.0836·101-s − 2.37·109-s − 1.81·121-s + 1.52·169-s − 3.27·171-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 160000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160000 ^{s/2} \, \Gamma_{\C}(s+7/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.989699010\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.989699010\) |
| \(L(\frac{9}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 + 355 p^{2} T^{2} + p^{14} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 218870 T^{2} + p^{14} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 1083 T + p^{7} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 95598010 T^{2} + p^{14} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 182154985 T^{2} + p^{14} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 33485 T + p^{7} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 6775568650 T^{2} + p^{14} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 4320 p T + p^{7} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 73798 T + p^{7} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 33106356790 T^{2} + p^{14} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 22683 T + p^{7} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 533607600310 T^{2} + p^{14} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 298337578610 T^{2} + p^{14} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 2223481045750 T^{2} + p^{14} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 1098360 T + p^{7} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 422998 T + p^{7} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 5575096711405 T^{2} + p^{14} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 2287428 T + p^{7} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 18513232750055 T^{2} + p^{14} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 2019250 T + p^{7} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 9296355939035 T^{2} + p^{14} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 2185935 T + p^{7} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 127681716222910 T^{2} + p^{14} T^{4} \) |
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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
−10.28553245200252350542648306989, −9.716196151371406918058975460648, −9.405848097716041164373313428871, −9.057367182708115205388299275724, −8.507680748507274927343467448550, −7.88523346875732577059549409971, −7.79065733603483346280549068398, −6.91268398962356110599384772968, −6.74241617751098050671777558426, −5.88284743871691640006606437829, −5.48211200302360955661429513243, −5.34967531512983842585948735665, −4.52226438910117576806187432688, −3.83748449968901356913049260413, −3.33780238285647297009619064364, −2.91213774499349755205036508504, −2.29902490775446736144139792473, −1.54837192892290523831114577077, −0.942687678881445819547746059229, −0.33506945471669694365081234170,
0.33506945471669694365081234170, 0.942687678881445819547746059229, 1.54837192892290523831114577077, 2.29902490775446736144139792473, 2.91213774499349755205036508504, 3.33780238285647297009619064364, 3.83748449968901356913049260413, 4.52226438910117576806187432688, 5.34967531512983842585948735665, 5.48211200302360955661429513243, 5.88284743871691640006606437829, 6.74241617751098050671777558426, 6.91268398962356110599384772968, 7.79065733603483346280549068398, 7.88523346875732577059549409971, 8.507680748507274927343467448550, 9.057367182708115205388299275724, 9.405848097716041164373313428871, 9.716196151371406918058975460648, 10.28553245200252350542648306989
