| L(s) = 1 | + 2.17e15·2-s + 6.93e23·3-s + 2.21e30·4-s − 2.56e35·5-s + 1.51e39·6-s + 8.01e42·7-s − 7.08e44·8-s − 1.06e48·9-s − 5.58e50·10-s − 9.55e51·11-s + 1.53e54·12-s + 4.33e54·13-s + 1.74e58·14-s − 1.77e59·15-s − 7.14e60·16-s − 7.76e61·17-s − 2.31e63·18-s + 3.75e64·19-s − 5.66e65·20-s + 5.56e66·21-s − 2.08e67·22-s − 1.01e69·23-s − 4.91e68·24-s + 2.62e70·25-s + 9.45e69·26-s − 1.81e72·27-s + 1.77e73·28-s + ⋯ |
| L(s) = 1 | + 1.36·2-s + 0.557·3-s + 0.871·4-s − 1.29·5-s + 0.763·6-s + 1.68·7-s − 0.175·8-s − 0.688·9-s − 1.76·10-s − 0.245·11-s + 0.486·12-s + 0.0241·13-s + 2.30·14-s − 0.719·15-s − 1.11·16-s − 0.565·17-s − 0.942·18-s + 0.994·19-s − 1.12·20-s + 0.939·21-s − 0.335·22-s − 1.74·23-s − 0.0979·24-s + 0.665·25-s + 0.0330·26-s − 0.942·27-s + 1.46·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & -\,\Lambda(102-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+50.5) \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(51)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{103}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| good | 2 | \( 1 - 2.17e15T + 2.53e30T^{2} \) |
| 3 | \( 1 - 6.93e23T + 1.54e48T^{2} \) |
| 5 | \( 1 + 2.56e35T + 3.94e70T^{2} \) |
| 7 | \( 1 - 8.01e42T + 2.26e85T^{2} \) |
| 11 | \( 1 + 9.55e51T + 1.51e105T^{2} \) |
| 13 | \( 1 - 4.33e54T + 3.22e112T^{2} \) |
| 17 | \( 1 + 7.76e61T + 1.88e124T^{2} \) |
| 19 | \( 1 - 3.75e64T + 1.42e129T^{2} \) |
| 23 | \( 1 + 1.01e69T + 3.42e137T^{2} \) |
| 29 | \( 1 + 3.74e73T + 5.03e147T^{2} \) |
| 31 | \( 1 + 5.68e74T + 4.24e150T^{2} \) |
| 37 | \( 1 + 2.65e79T + 2.44e158T^{2} \) |
| 41 | \( 1 - 3.81e81T + 7.78e162T^{2} \) |
| 43 | \( 1 + 3.91e82T + 9.55e164T^{2} \) |
| 47 | \( 1 - 2.04e83T + 7.61e168T^{2} \) |
| 53 | \( 1 + 5.06e86T + 1.41e174T^{2} \) |
| 59 | \( 1 - 5.27e89T + 7.17e178T^{2} \) |
| 61 | \( 1 + 2.08e90T + 2.08e180T^{2} \) |
| 67 | \( 1 + 4.63e91T + 2.71e184T^{2} \) |
| 71 | \( 1 + 3.91e93T + 9.48e186T^{2} \) |
| 73 | \( 1 - 1.42e94T + 1.56e188T^{2} \) |
| 79 | \( 1 + 5.87e95T + 4.57e191T^{2} \) |
| 83 | \( 1 - 6.23e96T + 6.71e193T^{2} \) |
| 89 | \( 1 + 4.22e98T + 7.73e196T^{2} \) |
| 97 | \( 1 - 1.04e100T + 4.61e200T^{2} \) |
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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
−13.83831344733983733217271901961, −11.97742793564205785717941294393, −11.29677745186060081272286594914, −8.558908603876670007740986979807, −7.58597400267031994504807971913, −5.49914897161061994479075920339, −4.38574778047642919774093686592, −3.43603241584958744486174212455, −2.03644198360498200132593859739, 0,
2.03644198360498200132593859739, 3.43603241584958744486174212455, 4.38574778047642919774093686592, 5.49914897161061994479075920339, 7.58597400267031994504807971913, 8.558908603876670007740986979807, 11.29677745186060081272286594914, 11.97742793564205785717941294393, 13.83831344733983733217271901961
