| L(s) = 1 | − 2.51·2-s + 4.33·4-s − 1.58·5-s + 3.57·7-s − 5.87·8-s + 3.97·10-s − 6.04·11-s − 5.34·13-s − 8.98·14-s + 6.11·16-s − 0.189·17-s − 19-s − 6.85·20-s + 15.2·22-s − 7.77·23-s − 2.49·25-s + 13.4·26-s + 15.4·28-s + 6.30·29-s − 7.81·31-s − 3.63·32-s + 0.477·34-s − 5.64·35-s − 7.85·37-s + 2.51·38-s + 9.28·40-s − 9.08·41-s + ⋯ |
| L(s) = 1 | − 1.77·2-s + 2.16·4-s − 0.707·5-s + 1.34·7-s − 2.07·8-s + 1.25·10-s − 1.82·11-s − 1.48·13-s − 2.40·14-s + 1.52·16-s − 0.0459·17-s − 0.229·19-s − 1.53·20-s + 3.24·22-s − 1.62·23-s − 0.499·25-s + 2.63·26-s + 2.92·28-s + 1.17·29-s − 1.40·31-s − 0.643·32-s + 0.0818·34-s − 0.954·35-s − 1.29·37-s + 0.408·38-s + 1.46·40-s − 1.41·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8037 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8037 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1184007838\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1184007838\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 19 | \( 1 + T \) |
| 47 | \( 1 + T \) |
| good | 2 | \( 1 + 2.51T + 2T^{2} \) |
| 5 | \( 1 + 1.58T + 5T^{2} \) |
| 7 | \( 1 - 3.57T + 7T^{2} \) |
| 11 | \( 1 + 6.04T + 11T^{2} \) |
| 13 | \( 1 + 5.34T + 13T^{2} \) |
| 17 | \( 1 + 0.189T + 17T^{2} \) |
| 23 | \( 1 + 7.77T + 23T^{2} \) |
| 29 | \( 1 - 6.30T + 29T^{2} \) |
| 31 | \( 1 + 7.81T + 31T^{2} \) |
| 37 | \( 1 + 7.85T + 37T^{2} \) |
| 41 | \( 1 + 9.08T + 41T^{2} \) |
| 43 | \( 1 + 0.672T + 43T^{2} \) |
| 53 | \( 1 - 10.0T + 53T^{2} \) |
| 59 | \( 1 + 4.72T + 59T^{2} \) |
| 61 | \( 1 + 8.43T + 61T^{2} \) |
| 67 | \( 1 + 8.88T + 67T^{2} \) |
| 71 | \( 1 + 8.51T + 71T^{2} \) |
| 73 | \( 1 + 8.58T + 73T^{2} \) |
| 79 | \( 1 - 15.8T + 79T^{2} \) |
| 83 | \( 1 + 5.89T + 83T^{2} \) |
| 89 | \( 1 + 9.24T + 89T^{2} \) |
| 97 | \( 1 + 9.88T + 97T^{2} \) |
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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
−7.82409476178199041951000685024, −7.57536938089427325553836891265, −7.01298956057073316057883674136, −5.83918839687633046567350980112, −5.07210290829086322959307850069, −4.42710566670222237319358696241, −3.12253825086369322862061694639, −2.18754959102428439390150998278, −1.76384794188418268342231523495, −0.21079632101744018686170634748,
0.21079632101744018686170634748, 1.76384794188418268342231523495, 2.18754959102428439390150998278, 3.12253825086369322862061694639, 4.42710566670222237319358696241, 5.07210290829086322959307850069, 5.83918839687633046567350980112, 7.01298956057073316057883674136, 7.57536938089427325553836891265, 7.82409476178199041951000685024
