| L(s) = 1 | − 140·2-s + 1.14e4·4-s − 4.87e4·5-s + 4.87e5·7-s − 4.50e5·8-s + 6.82e6·10-s + 1.77e6·11-s − 1.83e7·13-s − 6.82e7·14-s − 3.04e7·16-s + 9.62e7·17-s − 1.49e7·19-s − 5.56e8·20-s − 2.48e8·22-s − 1.53e8·23-s + 1.15e9·25-s + 2.57e9·26-s + 5.56e9·28-s − 5.21e9·29-s + 1.18e9·31-s + 7.94e9·32-s − 1.34e10·34-s − 2.37e10·35-s − 1.76e10·37-s + 2.09e9·38-s + 2.19e10·40-s + 1.94e10·41-s + ⋯ |
| L(s) = 1 | − 1.54·2-s + 1.39·4-s − 1.39·5-s + 1.56·7-s − 0.607·8-s + 2.15·10-s + 0.301·11-s − 1.05·13-s − 2.42·14-s − 0.453·16-s + 0.966·17-s − 0.0729·19-s − 1.94·20-s − 0.466·22-s − 0.216·23-s + 0.946·25-s + 1.63·26-s + 2.18·28-s − 1.62·29-s + 0.239·31-s + 1.30·32-s − 1.49·34-s − 2.18·35-s − 1.13·37-s + 0.112·38-s + 0.847·40-s + 0.639·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(7)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{15}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 - p^{6} T \) |
| good | 2 | \( 1 + 35 p^{2} T + p^{13} T^{2} \) |
| 5 | \( 1 + 9748 p T + p^{13} T^{2} \) |
| 7 | \( 1 - 487486 T + p^{13} T^{2} \) |
| 13 | \( 1 + 18388304 T + p^{13} T^{2} \) |
| 17 | \( 1 - 96233254 T + p^{13} T^{2} \) |
| 19 | \( 1 + 14954652 T + p^{13} T^{2} \) |
| 23 | \( 1 + 153804394 T + p^{13} T^{2} \) |
| 29 | \( 1 + 5219010534 T + p^{13} T^{2} \) |
| 31 | \( 1 - 1183811728 T + p^{13} T^{2} \) |
| 37 | \( 1 + 17672200362 T + p^{13} T^{2} \) |
| 41 | \( 1 - 19461739306 T + p^{13} T^{2} \) |
| 43 | \( 1 + 79355928 p T + p^{13} T^{2} \) |
| 47 | \( 1 - 100327719050 T + p^{13} T^{2} \) |
| 53 | \( 1 + 275469097716 T + p^{13} T^{2} \) |
| 59 | \( 1 - 267676863080 T + p^{13} T^{2} \) |
| 61 | \( 1 - 563486626260 T + p^{13} T^{2} \) |
| 67 | \( 1 - 1080842815700 T + p^{13} T^{2} \) |
| 71 | \( 1 - 1150562265222 T + p^{13} T^{2} \) |
| 73 | \( 1 + 345914515454 T + p^{13} T^{2} \) |
| 79 | \( 1 + 2004080959294 T + p^{13} T^{2} \) |
| 83 | \( 1 - 3336732240564 T + p^{13} T^{2} \) |
| 89 | \( 1 - 5696238036294 T + p^{13} T^{2} \) |
| 97 | \( 1 + 6550114593202 T + p^{13} T^{2} \) |
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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
−10.83642149821306794272396642667, −9.607920743514258974019144385801, −8.429427844521166822799626178172, −7.78016025079190298318576420829, −7.20046986618296394972468690067, −5.11481480960545960599755526758, −3.91511648887501183612240997317, −2.13564075699387035323201718910, −1.03492819896378162697610157802, 0,
1.03492819896378162697610157802, 2.13564075699387035323201718910, 3.91511648887501183612240997317, 5.11481480960545960599755526758, 7.20046986618296394972468690067, 7.78016025079190298318576420829, 8.429427844521166822799626178172, 9.607920743514258974019144385801, 10.83642149821306794272396642667
