| L(s) = 1 | − 2.04e3·2-s + 2.29e5·3-s + 4.19e6·4-s + 4.88e7·5-s − 4.70e8·6-s + 1.11e9·7-s − 8.58e9·8-s − 4.14e10·9-s − 1.00e11·10-s − 9.54e11·11-s + 9.62e11·12-s − 2.42e12·13-s − 2.29e12·14-s + 1.12e13·15-s + 1.75e13·16-s − 1.35e14·17-s + 8.48e13·18-s + 8.28e14·19-s + 2.04e14·20-s + 2.56e14·21-s + 1.95e15·22-s − 7.88e13·23-s − 1.97e15·24-s + 2.38e15·25-s + 4.95e15·26-s − 3.11e16·27-s + 4.69e15·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.748·3-s + 0.5·4-s + 0.447·5-s − 0.528·6-s + 0.213·7-s − 0.353·8-s − 0.440·9-s − 0.316·10-s − 1.00·11-s + 0.374·12-s − 0.374·13-s − 0.151·14-s + 0.334·15-s + 0.250·16-s − 0.957·17-s + 0.311·18-s + 1.63·19-s + 0.223·20-s + 0.160·21-s + 0.712·22-s − 0.0172·23-s − 0.264·24-s + 0.200·25-s + 0.264·26-s − 1.07·27-s + 0.106·28-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 | \( 1 - 2.29e5T + 9.41e10T^{2} \) |
| 7 | \( 1 - 1.11e9T + 2.73e19T^{2} \) |
| 11 | \( 1 + 9.54e11T + 8.95e23T^{2} \) |
| 13 | \( 1 + 2.42e12T + 4.17e25T^{2} \) |
| 17 | \( 1 + 1.35e14T + 1.99e28T^{2} \) |
| 19 | \( 1 - 8.28e14T + 2.57e29T^{2} \) |
| 23 | \( 1 + 7.88e13T + 2.08e31T^{2} \) |
| 29 | \( 1 + 3.21e16T + 4.31e33T^{2} \) |
| 31 | \( 1 + 1.85e17T + 2.00e34T^{2} \) |
| 37 | \( 1 + 3.74e17T + 1.17e36T^{2} \) |
| 41 | \( 1 + 3.22e18T + 1.24e37T^{2} \) |
| 43 | \( 1 + 2.74e18T + 3.71e37T^{2} \) |
| 47 | \( 1 - 5.25e17T + 2.87e38T^{2} \) |
| 53 | \( 1 + 4.40e19T + 4.55e39T^{2} \) |
| 59 | \( 1 - 1.56e20T + 5.36e40T^{2} \) |
| 61 | \( 1 - 5.13e20T + 1.15e41T^{2} \) |
| 67 | \( 1 + 7.59e19T + 9.99e41T^{2} \) |
| 71 | \( 1 + 4.54e20T + 3.79e42T^{2} \) |
| 73 | \( 1 + 4.50e21T + 7.18e42T^{2} \) |
| 79 | \( 1 + 1.03e22T + 4.42e43T^{2} \) |
| 83 | \( 1 + 5.91e21T + 1.37e44T^{2} \) |
| 89 | \( 1 + 2.39e22T + 6.85e44T^{2} \) |
| 97 | \( 1 - 1.24e23T + 4.96e45T^{2} \) |
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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
−14.58169983205091940843880867931, −13.25939727512564272838476998147, −11.35150879129843082742683915823, −9.828036761207862126136814570754, −8.624538790419002043482374226643, −7.35627020794432526751101012849, −5.40606186739822459580152865245, −3.06197370869439677911210663828, −1.87873178612466061232391881201, 0,
1.87873178612466061232391881201, 3.06197370869439677911210663828, 5.40606186739822459580152865245, 7.35627020794432526751101012849, 8.624538790419002043482374226643, 9.828036761207862126136814570754, 11.35150879129843082742683915823, 13.25939727512564272838476998147, 14.58169983205091940843880867931
