| L(s) = 1 | + 12·2-s + 729·3-s − 8.04e3·4-s + 8.74e3·6-s − 2.35e5·7-s − 1.94e5·8-s + 5.31e5·9-s − 1.11e7·11-s − 5.86e6·12-s − 8.04e6·13-s − 2.82e6·14-s + 6.35e7·16-s + 1.17e8·17-s + 6.37e6·18-s − 2.14e8·19-s − 1.71e8·21-s − 1.34e8·22-s − 8.30e8·23-s − 1.42e8·24-s − 9.65e7·26-s + 3.87e8·27-s + 1.89e9·28-s − 1.25e9·29-s + 6.15e9·31-s + 2.35e9·32-s − 8.15e9·33-s + 1.40e9·34-s + ⋯ |
| L(s) = 1 | + 0.132·2-s + 0.577·3-s − 0.982·4-s + 0.0765·6-s − 0.755·7-s − 0.262·8-s + 1/3·9-s − 1.90·11-s − 0.567·12-s − 0.462·13-s − 0.100·14-s + 0.947·16-s + 1.18·17-s + 0.0441·18-s − 1.04·19-s − 0.436·21-s − 0.252·22-s − 1.16·23-s − 0.151·24-s − 0.0613·26-s + 0.192·27-s + 0.741·28-s − 0.390·29-s + 1.24·31-s + 0.388·32-s − 1.09·33-s + 0.156·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(7)\) |
\(\approx\) |
\(1.066242755\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.066242755\) |
| \(L(\frac{15}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - p^{6} T \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 - 3 p^{2} T + p^{13} T^{2} \) |
| 7 | \( 1 + 33584 p T + p^{13} T^{2} \) |
| 11 | \( 1 + 1016628 p T + p^{13} T^{2} \) |
| 13 | \( 1 + 8049614 T + p^{13} T^{2} \) |
| 17 | \( 1 - 117494622 T + p^{13} T^{2} \) |
| 19 | \( 1 + 214061380 T + p^{13} T^{2} \) |
| 23 | \( 1 + 830555544 T + p^{13} T^{2} \) |
| 29 | \( 1 + 1252400250 T + p^{13} T^{2} \) |
| 31 | \( 1 - 6159350552 T + p^{13} T^{2} \) |
| 37 | \( 1 - 5498191402 T + p^{13} T^{2} \) |
| 41 | \( 1 + 4678687878 T + p^{13} T^{2} \) |
| 43 | \( 1 + 7115013764 T + p^{13} T^{2} \) |
| 47 | \( 1 - 29528776992 T + p^{13} T^{2} \) |
| 53 | \( 1 - 204125042466 T + p^{13} T^{2} \) |
| 59 | \( 1 + 29909821020 T + p^{13} T^{2} \) |
| 61 | \( 1 + 134392006738 T + p^{13} T^{2} \) |
| 67 | \( 1 + 348518801948 T + p^{13} T^{2} \) |
| 71 | \( 1 - 1314335409192 T + p^{13} T^{2} \) |
| 73 | \( 1 - 1178875922326 T + p^{13} T^{2} \) |
| 79 | \( 1 + 1072420659640 T + p^{13} T^{2} \) |
| 83 | \( 1 + 1124025139644 T + p^{13} T^{2} \) |
| 89 | \( 1 - 2235610909530 T + p^{13} T^{2} \) |
| 97 | \( 1 - 14215257165502 T + p^{13} T^{2} \) |
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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
−12.32243894955642949153708532886, −10.30779843682636864681395611845, −9.804254613921349637902838214229, −8.435192007404809118737799817910, −7.66308512910425170158560704108, −5.91128484255943175061829890448, −4.75728959687575702353086003618, −3.48172659001884078310196086585, −2.40400945580788702150296170099, −0.47798136406020288441805043728,
0.47798136406020288441805043728, 2.40400945580788702150296170099, 3.48172659001884078310196086585, 4.75728959687575702353086003618, 5.91128484255943175061829890448, 7.66308512910425170158560704108, 8.435192007404809118737799817910, 9.804254613921349637902838214229, 10.30779843682636864681395611845, 12.32243894955642949153708532886
