| L(s) = 1 | − 5.15·2-s + 4.91·3-s + 18.5·4-s − 5·5-s − 25.3·6-s + 12.2·7-s − 54.5·8-s − 2.81·9-s + 25.7·10-s + 0.204·11-s + 91.4·12-s − 13·13-s − 63.0·14-s − 24.5·15-s + 132.·16-s + 116.·17-s + 14.5·18-s − 19.1·19-s − 92.9·20-s + 60.0·21-s − 1.05·22-s − 162.·23-s − 268.·24-s + 25·25-s + 67.0·26-s − 146.·27-s + 227.·28-s + ⋯ |
| L(s) = 1 | − 1.82·2-s + 0.946·3-s + 2.32·4-s − 0.447·5-s − 1.72·6-s + 0.659·7-s − 2.41·8-s − 0.104·9-s + 0.815·10-s + 0.00560·11-s + 2.19·12-s − 0.277·13-s − 1.20·14-s − 0.423·15-s + 2.07·16-s + 1.65·17-s + 0.189·18-s − 0.230·19-s − 1.03·20-s + 0.624·21-s − 0.0102·22-s − 1.47·23-s − 2.28·24-s + 0.200·25-s + 0.505·26-s − 1.04·27-s + 1.53·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 + 5T \) |
| 13 | \( 1 + 13T \) |
| 31 | \( 1 - 31T \) |
| good | 2 | \( 1 + 5.15T + 8T^{2} \) |
| 3 | \( 1 - 4.91T + 27T^{2} \) |
| 7 | \( 1 - 12.2T + 343T^{2} \) |
| 11 | \( 1 - 0.204T + 1.33e3T^{2} \) |
| 17 | \( 1 - 116.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 19.1T + 6.85e3T^{2} \) |
| 23 | \( 1 + 162.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 40.9T + 2.43e4T^{2} \) |
| 37 | \( 1 - 6.38T + 5.06e4T^{2} \) |
| 41 | \( 1 - 47.2T + 6.89e4T^{2} \) |
| 43 | \( 1 + 221.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 393.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 321.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 173.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 691.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 415.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 608.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.19e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 682.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 650.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 651.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.24e3T + 9.12e5T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.337744978983126664358291383400, −7.87132809845025489493821749344, −7.50444123639571310198022674002, −6.39527213327364267745629094372, −5.40550907413986819713020961968, −3.95127519446271446316379340324, −2.95634741945509753619501759978, −2.11541052730486038328816671340, −1.18455938015339898843110371371, 0,
1.18455938015339898843110371371, 2.11541052730486038328816671340, 2.95634741945509753619501759978, 3.95127519446271446316379340324, 5.40550907413986819713020961968, 6.39527213327364267745629094372, 7.50444123639571310198022674002, 7.87132809845025489493821749344, 8.337744978983126664358291383400
