| L(s) = 1 | − 1.77·3-s − 4.64·7-s + 0.145·9-s + 4.47·11-s + 5.73·13-s + 3.54·17-s − 1.23·19-s + 8.23·21-s − 1.09·23-s + 5.06·27-s + 4.61·29-s + 6·31-s − 7.93·33-s − 2.19·37-s − 10.1·39-s − 2.61·41-s − 3.96·43-s + 7.51·47-s + 14.5·49-s − 6.29·51-s + 11.4·53-s + 2.19·57-s − 12.4·59-s + 1.14·61-s − 0.677·63-s − 2.19·67-s + 1.94·69-s + ⋯ |
| L(s) = 1 | − 1.02·3-s − 1.75·7-s + 0.0486·9-s + 1.34·11-s + 1.59·13-s + 0.860·17-s − 0.283·19-s + 1.79·21-s − 0.228·23-s + 0.974·27-s + 0.857·29-s + 1.07·31-s − 1.38·33-s − 0.360·37-s − 1.63·39-s − 0.408·41-s − 0.604·43-s + 1.09·47-s + 2.08·49-s − 0.881·51-s + 1.57·53-s + 0.290·57-s − 1.62·59-s + 0.146·61-s − 0.0853·63-s − 0.267·67-s + 0.234·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 500 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8831354941\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8831354941\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + 1.77T + 3T^{2} \) |
| 7 | \( 1 + 4.64T + 7T^{2} \) |
| 11 | \( 1 - 4.47T + 11T^{2} \) |
| 13 | \( 1 - 5.73T + 13T^{2} \) |
| 17 | \( 1 - 3.54T + 17T^{2} \) |
| 19 | \( 1 + 1.23T + 19T^{2} \) |
| 23 | \( 1 + 1.09T + 23T^{2} \) |
| 29 | \( 1 - 4.61T + 29T^{2} \) |
| 31 | \( 1 - 6T + 31T^{2} \) |
| 37 | \( 1 + 2.19T + 37T^{2} \) |
| 41 | \( 1 + 2.61T + 41T^{2} \) |
| 43 | \( 1 + 3.96T + 43T^{2} \) |
| 47 | \( 1 - 7.51T + 47T^{2} \) |
| 53 | \( 1 - 11.4T + 53T^{2} \) |
| 59 | \( 1 + 12.4T + 59T^{2} \) |
| 61 | \( 1 - 1.14T + 61T^{2} \) |
| 67 | \( 1 + 2.19T + 67T^{2} \) |
| 71 | \( 1 - 5.23T + 71T^{2} \) |
| 73 | \( 1 - 5.73T + 73T^{2} \) |
| 79 | \( 1 - 9.70T + 79T^{2} \) |
| 83 | \( 1 - 6.83T + 83T^{2} \) |
| 89 | \( 1 + 11.3T + 89T^{2} \) |
| 97 | \( 1 - 5.73T + 97T^{2} \) |
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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
−10.89840123908253505732204012781, −10.16480880399925323815629783794, −9.254263382856145397850742050045, −8.405182381629727819107109295081, −6.73089051092818377283417567034, −6.37159028971232070961782367569, −5.63560433698794103765912101318, −4.06132487035272542243068775845, −3.16902667586164829743777425028, −0.925094865522730985444199518214,
0.925094865522730985444199518214, 3.16902667586164829743777425028, 4.06132487035272542243068775845, 5.63560433698794103765912101318, 6.37159028971232070961782367569, 6.73089051092818377283417567034, 8.405182381629727819107109295081, 9.254263382856145397850742050045, 10.16480880399925323815629783794, 10.89840123908253505732204012781
