| L(s) = 1 | + 8·2-s + 27·3-s + 64·4-s + 216·6-s − 391·7-s + 512·8-s + 729·9-s − 4.39e3·11-s + 1.72e3·12-s − 1.34e4·13-s − 3.12e3·14-s + 4.09e3·16-s − 7.68e3·17-s + 5.83e3·18-s − 1.37e4·19-s − 1.05e4·21-s − 3.51e4·22-s + 3.54e4·23-s + 1.38e4·24-s − 1.07e5·26-s + 1.96e4·27-s − 2.50e4·28-s − 1.57e5·29-s − 9.93e4·31-s + 3.27e4·32-s − 1.18e5·33-s − 6.14e4·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.430·7-s + 0.353·8-s + 1/3·9-s − 0.996·11-s + 0.288·12-s − 1.69·13-s − 0.304·14-s + 1/4·16-s − 0.379·17-s + 0.235·18-s − 0.458·19-s − 0.248·21-s − 0.704·22-s + 0.608·23-s + 0.204·24-s − 1.20·26-s + 0.192·27-s − 0.215·28-s − 1.19·29-s − 0.598·31-s + 0.176·32-s − 0.575·33-s − 0.268·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - p^{3} T \) |
| 3 | \( 1 - p^{3} T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 391 T + p^{7} T^{2} \) |
| 11 | \( 1 + 4398 T + p^{7} T^{2} \) |
| 13 | \( 1 + 13447 T + p^{7} T^{2} \) |
| 17 | \( 1 + 7686 T + p^{7} T^{2} \) |
| 19 | \( 1 + 13705 T + p^{7} T^{2} \) |
| 23 | \( 1 - 35478 T + p^{7} T^{2} \) |
| 29 | \( 1 + 5430 p T + p^{7} T^{2} \) |
| 31 | \( 1 + 99343 T + p^{7} T^{2} \) |
| 37 | \( 1 + 161926 T + p^{7} T^{2} \) |
| 41 | \( 1 - 521952 T + p^{7} T^{2} \) |
| 43 | \( 1 - 340973 T + p^{7} T^{2} \) |
| 47 | \( 1 + 50886 T + p^{7} T^{2} \) |
| 53 | \( 1 + 891132 T + p^{7} T^{2} \) |
| 59 | \( 1 + 1344210 T + p^{7} T^{2} \) |
| 61 | \( 1 - 3394127 T + p^{7} T^{2} \) |
| 67 | \( 1 + 2248951 T + p^{7} T^{2} \) |
| 71 | \( 1 - 2731872 T + p^{7} T^{2} \) |
| 73 | \( 1 + 5028622 T + p^{7} T^{2} \) |
| 79 | \( 1 - 1571480 T + p^{7} T^{2} \) |
| 83 | \( 1 + 7792962 T + p^{7} T^{2} \) |
| 89 | \( 1 + 5802240 T + p^{7} T^{2} \) |
| 97 | \( 1 + 2498311 T + p^{7} T^{2} \) |
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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
−11.25258841772625242200787624769, −10.17748336722637566584502320095, −9.201494523341263027758106765818, −7.77774472272720924015286335605, −6.98729722911055328112727009382, −5.51668565121613846202165143927, −4.44201189174248336926615464622, −3.03836250160917733438050963707, −2.12331544147576611804086231581, 0,
2.12331544147576611804086231581, 3.03836250160917733438050963707, 4.44201189174248336926615464622, 5.51668565121613846202165143927, 6.98729722911055328112727009382, 7.77774472272720924015286335605, 9.201494523341263027758106765818, 10.17748336722637566584502320095, 11.25258841772625242200787624769
