L(s) = 1 | + 21.2·2-s + 81.9·3-s + 322.·4-s − 351.·5-s + 1.73e3·6-s − 895.·7-s + 4.13e3·8-s + 4.53e3·9-s − 7.45e3·10-s − 3.47e3·11-s + 2.64e4·12-s − 1.02e4·13-s − 1.90e4·14-s − 2.88e4·15-s + 4.64e4·16-s − 2.88e3·17-s + 9.61e4·18-s + 4.06e4·19-s − 1.13e5·20-s − 7.33e4·21-s − 7.36e4·22-s + 1.72e4·23-s + 3.38e5·24-s + 4.53e4·25-s − 2.18e5·26-s + 1.92e5·27-s − 2.88e5·28-s + ⋯ |
L(s) = 1 | + 1.87·2-s + 1.75·3-s + 2.52·4-s − 1.25·5-s + 3.28·6-s − 0.986·7-s + 2.85·8-s + 2.07·9-s − 2.35·10-s − 0.786·11-s + 4.41·12-s − 1.29·13-s − 1.85·14-s − 2.20·15-s + 2.83·16-s − 0.142·17-s + 3.88·18-s + 1.36·19-s − 3.16·20-s − 1.72·21-s − 1.47·22-s + 0.295·23-s + 5.00·24-s + 0.580·25-s − 2.43·26-s + 1.87·27-s − 2.48·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(4)\) |
\(\approx\) |
\(6.614298976\) |
\(L(\frac12)\) |
\(\approx\) |
\(6.614298976\) |
\(L(\frac{9}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 37 | \( 1 + 5.06e4T \) |
good | 2 | \( 1 - 21.2T + 128T^{2} \) |
| 3 | \( 1 - 81.9T + 2.18e3T^{2} \) |
| 5 | \( 1 + 351.T + 7.81e4T^{2} \) |
| 7 | \( 1 + 895.T + 8.23e5T^{2} \) |
| 11 | \( 1 + 3.47e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 1.02e4T + 6.27e7T^{2} \) |
| 17 | \( 1 + 2.88e3T + 4.10e8T^{2} \) |
| 19 | \( 1 - 4.06e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 1.72e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 7.52e4T + 1.72e10T^{2} \) |
| 31 | \( 1 - 2.34e5T + 2.75e10T^{2} \) |
| 41 | \( 1 - 4.59e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 9.45e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 4.91e4T + 5.06e11T^{2} \) |
| 53 | \( 1 + 3.28e5T + 1.17e12T^{2} \) |
| 59 | \( 1 + 5.57e5T + 2.48e12T^{2} \) |
| 61 | \( 1 - 1.05e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 1.86e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 4.01e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 2.10e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 7.60e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 7.35e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 2.01e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 9.68e6T + 8.07e13T^{2} \) |
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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
−14.72490979091403345265056848016, −13.70994120567855863244473812157, −12.82677467349840793431465964232, −11.86218116528140353683941337864, −9.946374427800502121750120871793, −7.909360737868793086777941092982, −7.03169288196411789557367600964, −4.69006233743045664076635235280, −3.38441135977466307347827911875, −2.70446975359335587073669934009,
2.70446975359335587073669934009, 3.38441135977466307347827911875, 4.69006233743045664076635235280, 7.03169288196411789557367600964, 7.909360737868793086777941092982, 9.946374427800502121750120871793, 11.86218116528140353683941337864, 12.82677467349840793431465964232, 13.70994120567855863244473812157, 14.72490979091403345265056848016