| L(s) = 1 | − 2-s + 3-s + 4-s + 1.64·5-s − 6-s − 3.72·7-s − 8-s + 9-s − 1.64·10-s − 2.87·11-s + 12-s − 13-s + 3.72·14-s + 1.64·15-s + 16-s + 6.32·17-s − 18-s + 5.24·19-s + 1.64·20-s − 3.72·21-s + 2.87·22-s − 1.37·23-s − 24-s − 2.30·25-s + 26-s + 27-s − 3.72·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.733·5-s − 0.408·6-s − 1.40·7-s − 0.353·8-s + 0.333·9-s − 0.518·10-s − 0.866·11-s + 0.288·12-s − 0.277·13-s + 0.994·14-s + 0.423·15-s + 0.250·16-s + 1.53·17-s − 0.235·18-s + 1.20·19-s + 0.366·20-s − 0.811·21-s + 0.612·22-s − 0.285·23-s − 0.204·24-s − 0.461·25-s + 0.196·26-s + 0.192·27-s − 0.703·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.621379649\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.621379649\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 103 | \( 1 + T \) |
| good | 5 | \( 1 - 1.64T + 5T^{2} \) |
| 7 | \( 1 + 3.72T + 7T^{2} \) |
| 11 | \( 1 + 2.87T + 11T^{2} \) |
| 17 | \( 1 - 6.32T + 17T^{2} \) |
| 19 | \( 1 - 5.24T + 19T^{2} \) |
| 23 | \( 1 + 1.37T + 23T^{2} \) |
| 29 | \( 1 + 3.11T + 29T^{2} \) |
| 31 | \( 1 - 5.73T + 31T^{2} \) |
| 37 | \( 1 - 6.24T + 37T^{2} \) |
| 41 | \( 1 - 1.98T + 41T^{2} \) |
| 43 | \( 1 + 4.35T + 43T^{2} \) |
| 47 | \( 1 - 6.51T + 47T^{2} \) |
| 53 | \( 1 + 2.20T + 53T^{2} \) |
| 59 | \( 1 + 0.319T + 59T^{2} \) |
| 61 | \( 1 + 8.24T + 61T^{2} \) |
| 67 | \( 1 - 13.3T + 67T^{2} \) |
| 71 | \( 1 - 0.644T + 71T^{2} \) |
| 73 | \( 1 + 12.8T + 73T^{2} \) |
| 79 | \( 1 + 0.103T + 79T^{2} \) |
| 83 | \( 1 + 6.99T + 83T^{2} \) |
| 89 | \( 1 + 4.93T + 89T^{2} \) |
| 97 | \( 1 + 16.9T + 97T^{2} \) |
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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
−7.75731292791659768311013599257, −7.39365172150209014703366383131, −6.52111863246390716603439805122, −5.80616895322484284226469649567, −5.35484465293468665584285130966, −4.07610770582453402977184363581, −3.00789753476131317502594170373, −2.88459899006678197136802255588, −1.75766137871363625548319713699, −0.67390001192917591410026029811,
0.67390001192917591410026029811, 1.75766137871363625548319713699, 2.88459899006678197136802255588, 3.00789753476131317502594170373, 4.07610770582453402977184363581, 5.35484465293468665584285130966, 5.80616895322484284226469649567, 6.52111863246390716603439805122, 7.39365172150209014703366383131, 7.75731292791659768311013599257
