| L(s) = 1 | − 8·2-s − 27·3-s + 64·4-s − 551.·5-s + 216·6-s − 627.·7-s − 512·8-s + 729·9-s + 4.41e3·10-s + 1.46e3·11-s − 1.72e3·12-s − 4.95e3·13-s + 5.02e3·14-s + 1.49e4·15-s + 4.09e3·16-s − 1.59e4·17-s − 5.83e3·18-s − 4.54e4·19-s − 3.53e4·20-s + 1.69e4·21-s − 1.16e4·22-s − 4.96e4·23-s + 1.38e4·24-s + 2.26e5·25-s + 3.96e4·26-s − 1.96e4·27-s − 4.01e4·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s − 1.97·5-s + 0.408·6-s − 0.691·7-s − 0.353·8-s + 0.333·9-s + 1.39·10-s + 0.330·11-s − 0.288·12-s − 0.625·13-s + 0.489·14-s + 1.14·15-s + 0.250·16-s − 0.785·17-s − 0.235·18-s − 1.52·19-s − 0.987·20-s + 0.399·21-s − 0.233·22-s − 0.851·23-s + 0.204·24-s + 2.89·25-s + 0.442·26-s − 0.192·27-s − 0.345·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 8T \) |
| 3 | \( 1 + 27T \) |
| 59 | \( 1 - 2.05e5T \) |
| good | 5 | \( 1 + 551.T + 7.81e4T^{2} \) |
| 7 | \( 1 + 627.T + 8.23e5T^{2} \) |
| 11 | \( 1 - 1.46e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 4.95e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 1.59e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 4.54e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 4.96e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 1.89e5T + 1.72e10T^{2} \) |
| 31 | \( 1 - 1.41e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 1.54e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 8.12e4T + 1.94e11T^{2} \) |
| 43 | \( 1 - 2.58e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 2.39e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 1.09e6T + 1.17e12T^{2} \) |
| 61 | \( 1 - 1.21e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 4.51e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 4.35e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 5.35e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 1.66e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 3.04e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 1.24e7T + 4.42e13T^{2} \) |
| 97 | \( 1 - 1.50e7T + 8.07e13T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.929745988983496625694935879748, −8.664688562285019146736436197341, −8.039007672909825638839759223367, −6.95786385658569626425184835759, −6.40216757974547264595570743298, −4.62955767053655787909343997183, −3.88540142176927653257362579434, −2.55694692851342445000418103114, −0.72213346019461211239752453675, 0,
0.72213346019461211239752453675, 2.55694692851342445000418103114, 3.88540142176927653257362579434, 4.62955767053655787909343997183, 6.40216757974547264595570743298, 6.95786385658569626425184835759, 8.039007672909825638839759223367, 8.664688562285019146736436197341, 9.929745988983496625694935879748
