| L(s) = 1 | − 4.09e3·2-s + 1.37e6·3-s + 1.67e7·4-s − 5.64e9·6-s − 3.00e10·7-s − 6.87e10·8-s + 1.05e12·9-s + 5.73e12·11-s + 2.31e13·12-s + 1.07e13·13-s + 1.23e14·14-s + 2.81e14·16-s − 2.97e15·17-s − 4.30e15·18-s − 5.42e15·19-s − 4.13e16·21-s − 2.34e16·22-s + 1.04e17·23-s − 9.46e16·24-s − 4.39e16·26-s + 2.81e17·27-s − 5.03e17·28-s + 3.09e18·29-s − 4.26e18·31-s − 1.15e18·32-s + 7.90e18·33-s + 1.21e19·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.49·3-s + 0.5·4-s − 1.05·6-s − 0.820·7-s − 0.353·8-s + 1.24·9-s + 0.551·11-s + 0.748·12-s + 0.127·13-s + 0.580·14-s + 0.250·16-s − 1.23·17-s − 0.877·18-s − 0.562·19-s − 1.22·21-s − 0.389·22-s + 0.994·23-s − 0.529·24-s − 0.0903·26-s + 0.361·27-s − 0.410·28-s + 1.62·29-s − 0.973·31-s − 0.176·32-s + 0.824·33-s + 0.875·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(26-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s+25/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + 4.09e3T \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 1.37e6T + 8.47e11T^{2} \) |
| 7 | \( 1 + 3.00e10T + 1.34e21T^{2} \) |
| 11 | \( 1 - 5.73e12T + 1.08e26T^{2} \) |
| 13 | \( 1 - 1.07e13T + 7.05e27T^{2} \) |
| 17 | \( 1 + 2.97e15T + 5.77e30T^{2} \) |
| 19 | \( 1 + 5.42e15T + 9.30e31T^{2} \) |
| 23 | \( 1 - 1.04e17T + 1.10e34T^{2} \) |
| 29 | \( 1 - 3.09e18T + 3.63e36T^{2} \) |
| 31 | \( 1 + 4.26e18T + 1.92e37T^{2} \) |
| 37 | \( 1 - 4.51e19T + 1.60e39T^{2} \) |
| 41 | \( 1 + 7.56e19T + 2.08e40T^{2} \) |
| 43 | \( 1 - 1.39e20T + 6.86e40T^{2} \) |
| 47 | \( 1 + 4.67e20T + 6.34e41T^{2} \) |
| 53 | \( 1 - 1.24e21T + 1.27e43T^{2} \) |
| 59 | \( 1 + 1.32e22T + 1.86e44T^{2} \) |
| 61 | \( 1 + 2.36e20T + 4.29e44T^{2} \) |
| 67 | \( 1 - 5.36e22T + 4.48e45T^{2} \) |
| 71 | \( 1 + 2.27e23T + 1.91e46T^{2} \) |
| 73 | \( 1 + 2.78e23T + 3.82e46T^{2} \) |
| 79 | \( 1 - 6.36e23T + 2.75e47T^{2} \) |
| 83 | \( 1 + 1.19e24T + 9.48e47T^{2} \) |
| 89 | \( 1 + 3.22e24T + 5.42e48T^{2} \) |
| 97 | \( 1 + 3.58e24T + 4.66e49T^{2} \) |
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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
−9.902187978979255746994516159569, −9.039714686875459904781508068143, −8.471425449580224367202345779404, −7.20873470411525535635754624127, −6.33209612018170452321617318316, −4.33939269053761495112626053445, −3.19169733405070615504727499730, −2.44808228372796036928895478094, −1.34046685561010609370492638404, 0,
1.34046685561010609370492638404, 2.44808228372796036928895478094, 3.19169733405070615504727499730, 4.33939269053761495112626053445, 6.33209612018170452321617318316, 7.20873470411525535635754624127, 8.471425449580224367202345779404, 9.039714686875459904781508068143, 9.902187978979255746994516159569
