L(s) = 1 | + 4.10·2-s − 3·3-s + 8.86·4-s − 12.3·6-s − 7·7-s + 3.54·8-s + 9·9-s − 8.17·11-s − 26.5·12-s + 19.2·13-s − 28.7·14-s − 56.3·16-s + 18.8·17-s + 36.9·18-s − 76.5·19-s + 21·21-s − 33.5·22-s − 142.·23-s − 10.6·24-s + 78.9·26-s − 27·27-s − 62.0·28-s − 96.1·29-s − 270.·31-s − 259.·32-s + 24.5·33-s + 77.6·34-s + ⋯ |
L(s) = 1 | + 1.45·2-s − 0.577·3-s + 1.10·4-s − 0.838·6-s − 0.377·7-s + 0.156·8-s + 0.333·9-s − 0.224·11-s − 0.639·12-s + 0.410·13-s − 0.548·14-s − 0.880·16-s + 0.269·17-s + 0.483·18-s − 0.923·19-s + 0.218·21-s − 0.325·22-s − 1.29·23-s − 0.0904·24-s + 0.595·26-s − 0.192·27-s − 0.418·28-s − 0.615·29-s − 1.56·31-s − 1.43·32-s + 0.129·33-s + 0.391·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + 3T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + 7T \) |
good | 2 | \( 1 - 4.10T + 8T^{2} \) |
| 11 | \( 1 + 8.17T + 1.33e3T^{2} \) |
| 13 | \( 1 - 19.2T + 2.19e3T^{2} \) |
| 17 | \( 1 - 18.8T + 4.91e3T^{2} \) |
| 19 | \( 1 + 76.5T + 6.85e3T^{2} \) |
| 23 | \( 1 + 142.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 96.1T + 2.43e4T^{2} \) |
| 31 | \( 1 + 270.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 335.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 122.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 492.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 96.9T + 1.03e5T^{2} \) |
| 53 | \( 1 + 388.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 112.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 347.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 101.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 304.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 753.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.16e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 889.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 938.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.20e3T + 9.12e5T^{2} \) |
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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
−10.27721490258472593846334379084, −9.246717001043376243220328163560, −8.001346642867866645583320358348, −6.78016870199804352023923756926, −6.05162157345321920998547532298, −5.31710995482067159098509395658, −4.24599064855875639307538210516, −3.44518004906898793119148682275, −2.03701401941646774248962406287, 0,
2.03701401941646774248962406287, 3.44518004906898793119148682275, 4.24599064855875639307538210516, 5.31710995482067159098509395658, 6.05162157345321920998547532298, 6.78016870199804352023923756926, 8.001346642867866645583320358348, 9.246717001043376243220328163560, 10.27721490258472593846334379084