| L(s) = 1 | + 2.04e3·2-s + 2.47e5·3-s + 4.19e6·4-s − 4.88e7·5-s + 5.06e8·6-s − 3.52e9·7-s + 8.58e9·8-s − 3.30e10·9-s − 1.00e11·10-s − 6.19e11·11-s + 1.03e12·12-s − 9.21e12·13-s − 7.22e12·14-s − 1.20e13·15-s + 1.75e13·16-s + 2.96e13·17-s − 6.76e13·18-s − 3.56e14·19-s − 2.04e14·20-s − 8.71e14·21-s − 1.26e15·22-s + 2.73e15·23-s + 2.12e15·24-s + 2.38e15·25-s − 1.88e16·26-s − 3.14e16·27-s − 1.47e16·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.805·3-s + 0.5·4-s − 0.447·5-s + 0.569·6-s − 0.674·7-s + 0.353·8-s − 0.350·9-s − 0.316·10-s − 0.654·11-s + 0.402·12-s − 1.42·13-s − 0.476·14-s − 0.360·15-s + 0.250·16-s + 0.209·17-s − 0.247·18-s − 0.701·19-s − 0.223·20-s − 0.543·21-s − 0.462·22-s + 0.597·23-s + 0.284·24-s + 0.200·25-s − 1.00·26-s − 1.08·27-s − 0.337·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(24-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s+23/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - 2.04e3T \) |
| 5 | \( 1 + 4.88e7T \) |
| good | 3 | \( 1 - 2.47e5T + 9.41e10T^{2} \) |
| 7 | \( 1 + 3.52e9T + 2.73e19T^{2} \) |
| 11 | \( 1 + 6.19e11T + 8.95e23T^{2} \) |
| 13 | \( 1 + 9.21e12T + 4.17e25T^{2} \) |
| 17 | \( 1 - 2.96e13T + 1.99e28T^{2} \) |
| 19 | \( 1 + 3.56e14T + 2.57e29T^{2} \) |
| 23 | \( 1 - 2.73e15T + 2.08e31T^{2} \) |
| 29 | \( 1 - 5.96e16T + 4.31e33T^{2} \) |
| 31 | \( 1 - 5.61e16T + 2.00e34T^{2} \) |
| 37 | \( 1 + 1.59e18T + 1.17e36T^{2} \) |
| 41 | \( 1 + 4.81e18T + 1.24e37T^{2} \) |
| 43 | \( 1 + 7.53e18T + 3.71e37T^{2} \) |
| 47 | \( 1 - 1.70e18T + 2.87e38T^{2} \) |
| 53 | \( 1 - 7.71e19T + 4.55e39T^{2} \) |
| 59 | \( 1 - 2.22e20T + 5.36e40T^{2} \) |
| 61 | \( 1 - 2.72e20T + 1.15e41T^{2} \) |
| 67 | \( 1 - 6.39e20T + 9.99e41T^{2} \) |
| 71 | \( 1 + 2.68e21T + 3.79e42T^{2} \) |
| 73 | \( 1 - 4.36e21T + 7.18e42T^{2} \) |
| 79 | \( 1 + 3.18e21T + 4.42e43T^{2} \) |
| 83 | \( 1 + 2.75e21T + 1.37e44T^{2} \) |
| 89 | \( 1 + 1.40e22T + 6.85e44T^{2} \) |
| 97 | \( 1 - 1.12e23T + 4.96e45T^{2} \) |
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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
−14.52714145053782251705922784833, −13.20883757430877632738312681175, −11.95917950019260127343796694263, −10.13369762318517327652199063516, −8.392302418733245324373918091481, −6.94103249727586938170408291690, −5.06211035121655743552037750147, −3.39131263719406303899277868045, −2.39426470800482057139001456287, 0,
2.39426470800482057139001456287, 3.39131263719406303899277868045, 5.06211035121655743552037750147, 6.94103249727586938170408291690, 8.392302418733245324373918091481, 10.13369762318517327652199063516, 11.95917950019260127343796694263, 13.20883757430877632738312681175, 14.52714145053782251705922784833
