| L(s) = 1 | − 2.37·2-s − 2.37·4-s − 5·5-s + 7.37·7-s + 24.6·8-s + 11.8·10-s + 4.13·11-s − 78.9·13-s − 17.4·14-s − 39.3·16-s + 33.3·17-s + 89.3·19-s + 11.8·20-s − 9.81·22-s + 199.·23-s + 25·25-s + 187.·26-s − 17.4·28-s − 50.4·29-s − 6·31-s − 103.·32-s − 79.0·34-s − 36.8·35-s − 290.·37-s − 211.·38-s − 123.·40-s + 53.3·41-s + ⋯ |
| L(s) = 1 | − 0.838·2-s − 0.296·4-s − 0.447·5-s + 0.398·7-s + 1.08·8-s + 0.375·10-s + 0.113·11-s − 1.68·13-s − 0.333·14-s − 0.615·16-s + 0.475·17-s + 1.07·19-s + 0.132·20-s − 0.0951·22-s + 1.80·23-s + 0.200·25-s + 1.41·26-s − 0.118·28-s − 0.323·29-s − 0.0347·31-s − 0.571·32-s − 0.398·34-s − 0.178·35-s − 1.28·37-s − 0.904·38-s − 0.486·40-s + 0.203·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + 5T \) |
| good | 2 | \( 1 + 2.37T + 8T^{2} \) |
| 7 | \( 1 - 7.37T + 343T^{2} \) |
| 11 | \( 1 - 4.13T + 1.33e3T^{2} \) |
| 13 | \( 1 + 78.9T + 2.19e3T^{2} \) |
| 17 | \( 1 - 33.3T + 4.91e3T^{2} \) |
| 19 | \( 1 - 89.3T + 6.85e3T^{2} \) |
| 23 | \( 1 - 199.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 50.4T + 2.43e4T^{2} \) |
| 31 | \( 1 + 6T + 2.97e4T^{2} \) |
| 37 | \( 1 + 290.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 53.3T + 6.89e4T^{2} \) |
| 43 | \( 1 + 298.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 418.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 399.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 98.2T + 2.05e5T^{2} \) |
| 61 | \( 1 + 683.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 225.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 512.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 994.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 201.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.10e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 372.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 139.T + 9.12e5T^{2} \) |
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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
−10.18177171908225922826524712617, −9.431624150473441886638381630360, −8.641444849449790634356388908158, −7.55055389687506104205181752667, −7.17254545231116624289259380952, −5.27567504975223743652264756551, −4.60444748632254013808338943803, −3.09630354773797817563931502604, −1.38462346633516835618027143510, 0,
1.38462346633516835618027143510, 3.09630354773797817563931502604, 4.60444748632254013808338943803, 5.27567504975223743652264756551, 7.17254545231116624289259380952, 7.55055389687506104205181752667, 8.641444849449790634356388908158, 9.431624150473441886638381630360, 10.18177171908225922826524712617
