| L(s) = 1 | + 2.51e3·2-s − 3.85e5·3-s − 2.07e6·4-s − 7.20e7·5-s − 9.68e8·6-s − 2.62e10·8-s + 5.43e10·9-s − 1.80e11·10-s + 9.26e11·11-s + 7.97e11·12-s − 5.15e12·13-s + 2.77e13·15-s − 4.87e13·16-s − 2.45e13·17-s + 1.36e14·18-s − 5.32e14·19-s + 1.49e14·20-s + 2.32e15·22-s + 1.17e15·23-s + 1.01e16·24-s − 6.73e15·25-s − 1.29e16·26-s + 1.53e16·27-s + 6.90e16·29-s + 6.97e16·30-s + 2.78e17·31-s + 9.80e16·32-s + ⋯ |
| L(s) = 1 | + 0.867·2-s − 1.25·3-s − 0.246·4-s − 0.659·5-s − 1.08·6-s − 1.08·8-s + 0.577·9-s − 0.572·10-s + 0.979·11-s + 0.310·12-s − 0.798·13-s + 0.828·15-s − 0.692·16-s − 0.173·17-s + 0.501·18-s − 1.04·19-s + 0.162·20-s + 0.849·22-s + 0.257·23-s + 1.35·24-s − 0.565·25-s − 0.692·26-s + 0.530·27-s + 1.05·29-s + 0.718·30-s + 1.96·31-s + 0.481·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(24-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+23/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(12)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{25}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| good | 2 | \( 1 - 2.51e3T + 8.38e6T^{2} \) |
| 3 | \( 1 + 3.85e5T + 9.41e10T^{2} \) |
| 5 | \( 1 + 7.20e7T + 1.19e16T^{2} \) |
| 11 | \( 1 - 9.26e11T + 8.95e23T^{2} \) |
| 13 | \( 1 + 5.15e12T + 4.17e25T^{2} \) |
| 17 | \( 1 + 2.45e13T + 1.99e28T^{2} \) |
| 19 | \( 1 + 5.32e14T + 2.57e29T^{2} \) |
| 23 | \( 1 - 1.17e15T + 2.08e31T^{2} \) |
| 29 | \( 1 - 6.90e16T + 4.31e33T^{2} \) |
| 31 | \( 1 - 2.78e17T + 2.00e34T^{2} \) |
| 37 | \( 1 - 1.43e18T + 1.17e36T^{2} \) |
| 41 | \( 1 + 1.70e18T + 1.24e37T^{2} \) |
| 43 | \( 1 - 6.33e18T + 3.71e37T^{2} \) |
| 47 | \( 1 - 1.35e19T + 2.87e38T^{2} \) |
| 53 | \( 1 + 7.01e19T + 4.55e39T^{2} \) |
| 59 | \( 1 + 2.39e20T + 5.36e40T^{2} \) |
| 61 | \( 1 + 1.37e20T + 1.15e41T^{2} \) |
| 67 | \( 1 - 6.78e20T + 9.99e41T^{2} \) |
| 71 | \( 1 - 2.14e21T + 3.79e42T^{2} \) |
| 73 | \( 1 + 3.07e21T + 7.18e42T^{2} \) |
| 79 | \( 1 - 6.81e21T + 4.42e43T^{2} \) |
| 83 | \( 1 + 8.36e21T + 1.37e44T^{2} \) |
| 89 | \( 1 + 3.41e22T + 6.85e44T^{2} \) |
| 97 | \( 1 - 1.21e22T + 4.96e45T^{2} \) |
| 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
−11.02469046953240326327632229800, −9.638514940729065200406667173904, −8.296668219731614662042684064793, −6.67743710322053578145398804669, −5.95196249684154952589333865173, −4.67672308715433555895708802778, −4.21483056241733635402268729052, −2.75642999150839468557593487390, −0.883424022329285193597900558143, 0,
0.883424022329285193597900558143, 2.75642999150839468557593487390, 4.21483056241733635402268729052, 4.67672308715433555895708802778, 5.95196249684154952589333865173, 6.67743710322053578145398804669, 8.296668219731614662042684064793, 9.638514940729065200406667173904, 11.02469046953240326327632229800
