| L(s) = 1 | − 1.67e7·2-s − 2.65e11·3-s + 2.81e14·4-s − 1.00e17·5-s + 4.46e18·6-s + 2.61e20·7-s − 4.72e21·8-s − 1.68e23·9-s + 1.68e24·10-s + 3.57e25·11-s − 7.48e25·12-s + 1.14e27·13-s − 4.39e27·14-s + 2.66e28·15-s + 7.92e28·16-s + 1.32e30·17-s + 2.82e30·18-s + 9.52e30·19-s − 2.82e31·20-s − 6.96e31·21-s − 6.00e32·22-s + 5.58e32·23-s + 1.25e33·24-s − 7.70e33·25-s − 1.91e34·26-s + 1.08e35·27-s + 7.37e34·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.543·3-s + 0.5·4-s − 0.752·5-s + 0.384·6-s + 0.516·7-s − 0.353·8-s − 0.704·9-s + 0.531·10-s + 1.09·11-s − 0.271·12-s + 0.584·13-s − 0.365·14-s + 0.408·15-s + 0.250·16-s + 0.949·17-s + 0.498·18-s + 0.446·19-s − 0.376·20-s − 0.280·21-s − 0.774·22-s + 0.242·23-s + 0.192·24-s − 0.434·25-s − 0.412·26-s + 0.926·27-s + 0.258·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(50-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2 ^{s/2} \, \Gamma_{\C}(s+49/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(25)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{51}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 1.67e7T \) |
| good | 3 | \( 1 + 2.65e11T + 2.39e23T^{2} \) |
| 5 | \( 1 + 1.00e17T + 1.77e34T^{2} \) |
| 7 | \( 1 - 2.61e20T + 2.56e41T^{2} \) |
| 11 | \( 1 - 3.57e25T + 1.06e51T^{2} \) |
| 13 | \( 1 - 1.14e27T + 3.83e54T^{2} \) |
| 17 | \( 1 - 1.32e30T + 1.95e60T^{2} \) |
| 19 | \( 1 - 9.52e30T + 4.55e62T^{2} \) |
| 23 | \( 1 - 5.58e32T + 5.30e66T^{2} \) |
| 29 | \( 1 + 1.10e36T + 4.54e71T^{2} \) |
| 31 | \( 1 + 5.00e36T + 1.19e73T^{2} \) |
| 37 | \( 1 + 4.70e38T + 6.94e76T^{2} \) |
| 41 | \( 1 - 1.94e39T + 1.06e79T^{2} \) |
| 43 | \( 1 - 1.05e40T + 1.09e80T^{2} \) |
| 47 | \( 1 - 7.27e40T + 8.56e81T^{2} \) |
| 53 | \( 1 + 1.56e42T + 3.08e84T^{2} \) |
| 59 | \( 1 - 3.97e43T + 5.91e86T^{2} \) |
| 61 | \( 1 + 3.57e43T + 3.02e87T^{2} \) |
| 67 | \( 1 + 9.28e44T + 3.00e89T^{2} \) |
| 71 | \( 1 + 3.02e45T + 5.14e90T^{2} \) |
| 73 | \( 1 + 2.38e45T + 2.00e91T^{2} \) |
| 79 | \( 1 - 1.79e46T + 9.63e92T^{2} \) |
| 83 | \( 1 + 1.36e47T + 1.08e94T^{2} \) |
| 89 | \( 1 - 8.51e47T + 3.31e95T^{2} \) |
| 97 | \( 1 - 4.37e48T + 2.24e97T^{2} \) |
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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
−16.32505129086880965855091484654, −14.56332069409854110068722960153, −11.95371952249700270094874721146, −11.03093205288358349022703629731, −8.975210981503701117147819781525, −7.47970327921472559890882513517, −5.70393353877030362855966441639, −3.61361412423275293026603192528, −1.40570036687269561729357193042, 0,
1.40570036687269561729357193042, 3.61361412423275293026603192528, 5.70393353877030362855966441639, 7.47970327921472559890882513517, 8.975210981503701117147819781525, 11.03093205288358349022703629731, 11.95371952249700270094874721146, 14.56332069409854110068722960153, 16.32505129086880965855091484654
