| L(s) = 1 | − 6.13e3·2-s − 1.60e6·3-s + 4.06e6·4-s − 2.44e8·5-s + 9.82e9·6-s − 2.17e9·7-s + 1.80e11·8-s + 1.71e12·9-s + 1.49e12·10-s − 8.98e12·11-s − 6.51e12·12-s + 1.16e14·13-s + 1.33e13·14-s + 3.91e14·15-s − 1.24e15·16-s + 2.27e15·17-s − 1.05e16·18-s − 1.75e16·19-s − 9.93e14·20-s + 3.48e15·21-s + 5.51e16·22-s + 1.45e15·23-s − 2.89e17·24-s + 5.96e16·25-s − 7.13e17·26-s − 1.39e18·27-s − 8.85e15·28-s + ⋯ |
| L(s) = 1 | − 1.05·2-s − 1.73·3-s + 0.121·4-s − 0.447·5-s + 1.84·6-s − 0.0594·7-s + 0.930·8-s + 2.02·9-s + 0.473·10-s − 0.863·11-s − 0.210·12-s + 1.38·13-s + 0.0629·14-s + 0.778·15-s − 1.10·16-s + 0.946·17-s − 2.14·18-s − 1.82·19-s − 0.0542·20-s + 0.103·21-s + 0.914·22-s + 0.0138·23-s − 1.61·24-s + 0.199·25-s − 1.46·26-s − 1.78·27-s − 0.00720·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(26-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s+25/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(13)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{27}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + 2.44e8T \) |
| good | 2 | \( 1 + 6.13e3T + 3.35e7T^{2} \) |
| 3 | \( 1 + 1.60e6T + 8.47e11T^{2} \) |
| 7 | \( 1 + 2.17e9T + 1.34e21T^{2} \) |
| 11 | \( 1 + 8.98e12T + 1.08e26T^{2} \) |
| 13 | \( 1 - 1.16e14T + 7.05e27T^{2} \) |
| 17 | \( 1 - 2.27e15T + 5.77e30T^{2} \) |
| 19 | \( 1 + 1.75e16T + 9.30e31T^{2} \) |
| 23 | \( 1 - 1.45e15T + 1.10e34T^{2} \) |
| 29 | \( 1 - 1.07e18T + 3.63e36T^{2} \) |
| 31 | \( 1 - 5.76e18T + 1.92e37T^{2} \) |
| 37 | \( 1 - 6.50e19T + 1.60e39T^{2} \) |
| 41 | \( 1 - 5.23e19T + 2.08e40T^{2} \) |
| 43 | \( 1 + 1.34e20T + 6.86e40T^{2} \) |
| 47 | \( 1 + 1.15e20T + 6.34e41T^{2} \) |
| 53 | \( 1 + 4.60e21T + 1.27e43T^{2} \) |
| 59 | \( 1 - 1.73e22T + 1.86e44T^{2} \) |
| 61 | \( 1 + 3.93e22T + 4.29e44T^{2} \) |
| 67 | \( 1 - 5.12e21T + 4.48e45T^{2} \) |
| 71 | \( 1 - 1.37e23T + 1.91e46T^{2} \) |
| 73 | \( 1 + 1.15e23T + 3.82e46T^{2} \) |
| 79 | \( 1 + 4.01e23T + 2.75e47T^{2} \) |
| 83 | \( 1 + 6.71e23T + 9.48e47T^{2} \) |
| 89 | \( 1 - 3.91e23T + 5.42e48T^{2} \) |
| 97 | \( 1 - 4.07e24T + 4.66e49T^{2} \) |
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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
−16.90023276290412527533256850681, −15.89389613869807719324648507440, −12.91165838172122825203476399450, −11.19710069747378123386744880405, −10.25091716954808457573763523643, −8.109784007818644316277920852861, −6.27363469757709022336066648207, −4.54120194123634480089516918435, −1.09332829874511815408889014654, 0,
1.09332829874511815408889014654, 4.54120194123634480089516918435, 6.27363469757709022336066648207, 8.109784007818644316277920852861, 10.25091716954808457573763523643, 11.19710069747378123386744880405, 12.91165838172122825203476399450, 15.89389613869807719324648507440, 16.90023276290412527533256850681
