| L(s) = 1 | + 4.09e3·2-s − 1.08e6·3-s + 1.67e7·4-s − 4.42e9·6-s − 5.43e10·7-s + 6.87e10·8-s + 3.19e11·9-s − 3.24e12·11-s − 1.81e13·12-s − 7.83e13·13-s − 2.22e14·14-s + 2.81e14·16-s + 4.33e15·17-s + 1.30e15·18-s + 8.87e15·19-s + 5.86e16·21-s − 1.32e16·22-s − 7.05e16·23-s − 7.42e16·24-s − 3.20e17·26-s + 5.70e17·27-s − 9.11e17·28-s − 1.11e18·29-s + 6.30e18·31-s + 1.15e18·32-s + 3.50e18·33-s + 1.77e19·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.17·3-s + 0.5·4-s − 0.829·6-s − 1.48·7-s + 0.353·8-s + 0.377·9-s − 0.311·11-s − 0.586·12-s − 0.932·13-s − 1.04·14-s + 0.250·16-s + 1.80·17-s + 0.266·18-s + 0.920·19-s + 1.74·21-s − 0.220·22-s − 0.671·23-s − 0.414·24-s − 0.659·26-s + 0.730·27-s − 0.741·28-s − 0.584·29-s + 1.43·31-s + 0.176·32-s + 0.365·33-s + 1.27·34-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 | \( 1 + 1.08e6T + 8.47e11T^{2} \) |
| 7 | \( 1 + 5.43e10T + 1.34e21T^{2} \) |
| 11 | \( 1 + 3.24e12T + 1.08e26T^{2} \) |
| 13 | \( 1 + 7.83e13T + 7.05e27T^{2} \) |
| 17 | \( 1 - 4.33e15T + 5.77e30T^{2} \) |
| 19 | \( 1 - 8.87e15T + 9.30e31T^{2} \) |
| 23 | \( 1 + 7.05e16T + 1.10e34T^{2} \) |
| 29 | \( 1 + 1.11e18T + 3.63e36T^{2} \) |
| 31 | \( 1 - 6.30e18T + 1.92e37T^{2} \) |
| 37 | \( 1 - 1.55e19T + 1.60e39T^{2} \) |
| 41 | \( 1 + 3.82e19T + 2.08e40T^{2} \) |
| 43 | \( 1 + 3.32e20T + 6.86e40T^{2} \) |
| 47 | \( 1 - 1.01e21T + 6.34e41T^{2} \) |
| 53 | \( 1 + 3.43e21T + 1.27e43T^{2} \) |
| 59 | \( 1 - 4.09e21T + 1.86e44T^{2} \) |
| 61 | \( 1 - 3.14e22T + 4.29e44T^{2} \) |
| 67 | \( 1 + 1.01e23T + 4.48e45T^{2} \) |
| 71 | \( 1 - 2.10e23T + 1.91e46T^{2} \) |
| 73 | \( 1 + 1.56e23T + 3.82e46T^{2} \) |
| 79 | \( 1 - 6.00e22T + 2.75e47T^{2} \) |
| 83 | \( 1 - 7.40e23T + 9.48e47T^{2} \) |
| 89 | \( 1 - 3.21e24T + 5.42e48T^{2} \) |
| 97 | \( 1 - 5.32e24T + 4.66e49T^{2} \) |
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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
−10.32465742370672425243997598061, −9.719815062990690276727364895368, −7.70831133431474565609551608163, −6.60461475651617123762411527485, −5.76338962881101524327713931909, −5.01999490427217786459971751307, −3.56045100734653545326331834112, −2.68459019158598909538897921551, −0.967074509945198751075113853278, 0,
0.967074509945198751075113853278, 2.68459019158598909538897921551, 3.56045100734653545326331834112, 5.01999490427217786459971751307, 5.76338962881101524327713931909, 6.60461475651617123762411527485, 7.70831133431474565609551608163, 9.719815062990690276727364895368, 10.32465742370672425243997598061
