| L(s) = 1 | + 1.03e16·2-s − 2.77e24·3-s + 6.60e31·4-s − 1.38e36·5-s − 2.86e40·6-s − 3.04e43·7-s + 2.63e47·8-s − 1.17e50·9-s − 1.43e52·10-s + 4.98e54·11-s − 1.83e56·12-s + 2.40e58·13-s − 3.14e59·14-s + 3.85e60·15-s + 4.01e61·16-s − 3.48e64·17-s − 1.21e66·18-s − 1.66e67·19-s − 9.17e67·20-s + 8.44e67·21-s + 5.14e70·22-s − 2.46e70·23-s − 7.30e71·24-s − 2.27e73·25-s + 2.48e74·26-s + 6.73e74·27-s − 2.01e75·28-s + ⋯ |
| L(s) = 1 | + 1.62·2-s − 0.247·3-s + 1.62·4-s − 0.279·5-s − 0.401·6-s − 0.130·7-s + 1.01·8-s − 0.938·9-s − 0.453·10-s + 1.05·11-s − 0.403·12-s + 0.791·13-s − 0.211·14-s + 0.0693·15-s + 0.0244·16-s − 0.879·17-s − 1.52·18-s − 1.22·19-s − 0.455·20-s + 0.0323·21-s + 1.71·22-s − 0.0795·23-s − 0.252·24-s − 0.921·25-s + 1.28·26-s + 0.480·27-s − 0.212·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & -\,\Lambda(106-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+52.5) \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(53)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{107}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| good | 2 | \( 1 - 1.03e16T + 4.05e31T^{2} \) |
| 3 | \( 1 + 2.77e24T + 1.25e50T^{2} \) |
| 5 | \( 1 + 1.38e36T + 2.46e73T^{2} \) |
| 7 | \( 1 + 3.04e43T + 5.43e88T^{2} \) |
| 11 | \( 1 - 4.98e54T + 2.21e109T^{2} \) |
| 13 | \( 1 - 2.40e58T + 9.20e116T^{2} \) |
| 17 | \( 1 + 3.48e64T + 1.57e129T^{2} \) |
| 19 | \( 1 + 1.66e67T + 1.85e134T^{2} \) |
| 23 | \( 1 + 2.46e70T + 9.58e142T^{2} \) |
| 29 | \( 1 + 7.76e76T + 3.56e153T^{2} \) |
| 31 | \( 1 - 1.96e78T + 3.91e156T^{2} \) |
| 37 | \( 1 + 3.69e82T + 4.58e164T^{2} \) |
| 41 | \( 1 + 4.85e84T + 2.19e169T^{2} \) |
| 43 | \( 1 - 9.79e85T + 3.26e171T^{2} \) |
| 47 | \( 1 + 3.79e87T + 3.71e175T^{2} \) |
| 53 | \( 1 - 1.90e90T + 1.11e181T^{2} \) |
| 59 | \( 1 + 2.84e92T + 8.69e185T^{2} \) |
| 61 | \( 1 - 3.64e93T + 2.88e187T^{2} \) |
| 67 | \( 1 + 4.96e95T + 5.46e191T^{2} \) |
| 71 | \( 1 - 1.33e97T + 2.41e194T^{2} \) |
| 73 | \( 1 + 8.83e97T + 4.45e195T^{2} \) |
| 79 | \( 1 + 1.15e99T + 1.78e199T^{2} \) |
| 83 | \( 1 + 5.00e100T + 3.18e201T^{2} \) |
| 89 | \( 1 - 6.61e101T + 4.85e204T^{2} \) |
| 97 | \( 1 - 1.82e104T + 4.08e208T^{2} \) |
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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
−13.34937577677660313719553505316, −11.96512864781676150786227663542, −11.07369036943310906136249180993, −8.732243789616008284186091035356, −6.65566115076316911901412442783, −5.79795191873918231946372269808, −4.34256204433287742239635233014, −3.42395226448728700774421044460, −1.96962397760348390164621000186, 0,
1.96962397760348390164621000186, 3.42395226448728700774421044460, 4.34256204433287742239635233014, 5.79795191873918231946372269808, 6.65566115076316911901412442783, 8.732243789616008284186091035356, 11.07369036943310906136249180993, 11.96512864781676150786227663542, 13.34937577677660313719553505316
