| L(s) = 1 | + 1.67e7·2-s + 7.94e11·3-s + 2.81e14·4-s − 5.65e16·5-s + 1.33e19·6-s − 4.38e20·7-s + 4.72e21·8-s + 3.91e23·9-s − 9.48e23·10-s + 5.68e25·11-s + 2.23e26·12-s + 1.61e27·13-s − 7.35e27·14-s − 4.49e28·15-s + 7.92e28·16-s + 2.39e30·17-s + 6.57e30·18-s − 9.22e30·19-s − 1.59e31·20-s − 3.48e32·21-s + 9.53e32·22-s − 1.45e33·23-s + 3.75e33·24-s − 1.45e34·25-s + 2.71e34·26-s + 1.21e35·27-s − 1.23e35·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.62·3-s + 0.5·4-s − 0.424·5-s + 1.14·6-s − 0.865·7-s + 0.353·8-s + 1.63·9-s − 0.299·10-s + 1.73·11-s + 0.811·12-s + 0.827·13-s − 0.611·14-s − 0.688·15-s + 0.250·16-s + 1.71·17-s + 1.15·18-s − 0.432·19-s − 0.212·20-s − 1.40·21-s + 1.22·22-s − 0.633·23-s + 0.574·24-s − 0.820·25-s + 0.584·26-s + 1.03·27-s − 0.432·28-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
| \(L(25)\) |
\(\approx\) |
\(5.337800343\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.337800343\) |
| \(L(\frac{51}{2})\) |
|
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 | \( 1 - 7.94e11T + 2.39e23T^{2} \) |
| 5 | \( 1 + 5.65e16T + 1.77e34T^{2} \) |
| 7 | \( 1 + 4.38e20T + 2.56e41T^{2} \) |
| 11 | \( 1 - 5.68e25T + 1.06e51T^{2} \) |
| 13 | \( 1 - 1.61e27T + 3.83e54T^{2} \) |
| 17 | \( 1 - 2.39e30T + 1.95e60T^{2} \) |
| 19 | \( 1 + 9.22e30T + 4.55e62T^{2} \) |
| 23 | \( 1 + 1.45e33T + 5.30e66T^{2} \) |
| 29 | \( 1 - 1.26e36T + 4.54e71T^{2} \) |
| 31 | \( 1 + 2.70e36T + 1.19e73T^{2} \) |
| 37 | \( 1 + 7.30e37T + 6.94e76T^{2} \) |
| 41 | \( 1 - 1.21e39T + 1.06e79T^{2} \) |
| 43 | \( 1 + 1.71e40T + 1.09e80T^{2} \) |
| 47 | \( 1 + 1.04e41T + 8.56e81T^{2} \) |
| 53 | \( 1 + 1.17e41T + 3.08e84T^{2} \) |
| 59 | \( 1 + 8.71e42T + 5.91e86T^{2} \) |
| 61 | \( 1 + 2.61e43T + 3.02e87T^{2} \) |
| 67 | \( 1 - 3.87e44T + 3.00e89T^{2} \) |
| 71 | \( 1 - 2.06e44T + 5.14e90T^{2} \) |
| 73 | \( 1 - 4.21e44T + 2.00e91T^{2} \) |
| 79 | \( 1 - 8.65e45T + 9.63e92T^{2} \) |
| 83 | \( 1 - 1.57e47T + 1.08e94T^{2} \) |
| 89 | \( 1 + 2.35e47T + 3.31e95T^{2} \) |
| 97 | \( 1 - 1.57e48T + 2.24e97T^{2} \) |
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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
−16.20456799217905896884768467306, −14.73896957976704895223707120740, −13.75322535842257672037216602098, −12.16424367654486949592547232511, −9.687572339335342823233636254144, −8.188336359086953354755005804921, −6.51203288617538966076572647104, −3.87923887090072069028369432781, −3.25043929444362311762323158344, −1.48282513455735403202035086127,
1.48282513455735403202035086127, 3.25043929444362311762323158344, 3.87923887090072069028369432781, 6.51203288617538966076572647104, 8.188336359086953354755005804921, 9.687572339335342823233636254144, 12.16424367654486949592547232511, 13.75322535842257672037216602098, 14.73896957976704895223707120740, 16.20456799217905896884768467306
