| L(s) = 1 | + 4.09e3·2-s − 2.33e5·3-s + 1.67e7·4-s − 9.57e8·6-s + 6.39e10·7-s + 6.87e10·8-s − 7.92e11·9-s + 6.04e12·11-s − 3.92e12·12-s − 1.13e14·13-s + 2.61e14·14-s + 2.81e14·16-s − 9.52e14·17-s − 3.24e15·18-s + 8.44e15·19-s − 1.49e16·21-s + 2.47e16·22-s + 1.51e16·23-s − 1.60e16·24-s − 4.66e17·26-s + 3.83e17·27-s + 1.07e18·28-s − 1.63e18·29-s − 7.82e18·31-s + 1.15e18·32-s − 1.41e18·33-s − 3.90e18·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.253·3-s + 0.5·4-s − 0.179·6-s + 1.74·7-s + 0.353·8-s − 0.935·9-s + 0.580·11-s − 0.126·12-s − 1.35·13-s + 1.23·14-s + 0.250·16-s − 0.396·17-s − 0.661·18-s + 0.875·19-s − 0.443·21-s + 0.410·22-s + 0.144·23-s − 0.0897·24-s − 0.958·26-s + 0.491·27-s + 0.872·28-s − 0.857·29-s − 1.78·31-s + 0.176·32-s − 0.147·33-s − 0.280·34-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 | \( 1 + 2.33e5T + 8.47e11T^{2} \) |
| 7 | \( 1 - 6.39e10T + 1.34e21T^{2} \) |
| 11 | \( 1 - 6.04e12T + 1.08e26T^{2} \) |
| 13 | \( 1 + 1.13e14T + 7.05e27T^{2} \) |
| 17 | \( 1 + 9.52e14T + 5.77e30T^{2} \) |
| 19 | \( 1 - 8.44e15T + 9.30e31T^{2} \) |
| 23 | \( 1 - 1.51e16T + 1.10e34T^{2} \) |
| 29 | \( 1 + 1.63e18T + 3.63e36T^{2} \) |
| 31 | \( 1 + 7.82e18T + 1.92e37T^{2} \) |
| 37 | \( 1 + 3.53e19T + 1.60e39T^{2} \) |
| 41 | \( 1 + 2.35e20T + 2.08e40T^{2} \) |
| 43 | \( 1 + 3.29e19T + 6.86e40T^{2} \) |
| 47 | \( 1 + 1.82e20T + 6.34e41T^{2} \) |
| 53 | \( 1 - 5.40e21T + 1.27e43T^{2} \) |
| 59 | \( 1 + 1.13e22T + 1.86e44T^{2} \) |
| 61 | \( 1 + 2.83e22T + 4.29e44T^{2} \) |
| 67 | \( 1 - 1.16e23T + 4.48e45T^{2} \) |
| 71 | \( 1 - 1.14e23T + 1.91e46T^{2} \) |
| 73 | \( 1 - 8.85e21T + 3.82e46T^{2} \) |
| 79 | \( 1 - 2.17e23T + 2.75e47T^{2} \) |
| 83 | \( 1 + 6.46e22T + 9.48e47T^{2} \) |
| 89 | \( 1 - 3.57e24T + 5.42e48T^{2} \) |
| 97 | \( 1 + 2.49e24T + 4.66e49T^{2} \) |
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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
−10.75324786836759117523821041632, −9.143634596112321483450180563252, −7.930620751788906991667412280543, −6.96390855042893177862218392042, −5.37874090193503905283651040129, −5.02093607160225073258093450480, −3.70218212295761472582509372544, −2.33431899642090755249656275690, −1.47087784794292522520309692647, 0,
1.47087784794292522520309692647, 2.33431899642090755249656275690, 3.70218212295761472582509372544, 5.02093607160225073258093450480, 5.37874090193503905283651040129, 6.96390855042893177862218392042, 7.930620751788906991667412280543, 9.143634596112321483450180563252, 10.75324786836759117523821041632
