| L(s) = 1 | + 0.563·2-s − 1.68·4-s + 1.59·5-s + 4.95·7-s − 2.07·8-s + 0.897·10-s + 0.616·11-s + 6.48·13-s + 2.78·14-s + 2.19·16-s + 1.67·17-s + 2.69·19-s − 2.68·20-s + 0.347·22-s + 23-s − 2.45·25-s + 3.65·26-s − 8.33·28-s + 29-s − 3.94·31-s + 5.38·32-s + 0.943·34-s + 7.89·35-s + 5.81·37-s + 1.51·38-s − 3.30·40-s + 6.01·41-s + ⋯ |
| L(s) = 1 | + 0.398·2-s − 0.841·4-s + 0.713·5-s + 1.87·7-s − 0.733·8-s + 0.283·10-s + 0.185·11-s + 1.79·13-s + 0.745·14-s + 0.549·16-s + 0.406·17-s + 0.618·19-s − 0.599·20-s + 0.0740·22-s + 0.208·23-s − 0.491·25-s + 0.716·26-s − 1.57·28-s + 0.185·29-s − 0.708·31-s + 0.952·32-s + 0.161·34-s + 1.33·35-s + 0.956·37-s + 0.246·38-s − 0.522·40-s + 0.938·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.436514131\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.436514131\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 - 0.563T + 2T^{2} \) |
| 5 | \( 1 - 1.59T + 5T^{2} \) |
| 7 | \( 1 - 4.95T + 7T^{2} \) |
| 11 | \( 1 - 0.616T + 11T^{2} \) |
| 13 | \( 1 - 6.48T + 13T^{2} \) |
| 17 | \( 1 - 1.67T + 17T^{2} \) |
| 19 | \( 1 - 2.69T + 19T^{2} \) |
| 31 | \( 1 + 3.94T + 31T^{2} \) |
| 37 | \( 1 - 5.81T + 37T^{2} \) |
| 41 | \( 1 - 6.01T + 41T^{2} \) |
| 43 | \( 1 - 0.472T + 43T^{2} \) |
| 47 | \( 1 + 4.27T + 47T^{2} \) |
| 53 | \( 1 - 7.82T + 53T^{2} \) |
| 59 | \( 1 + 11.6T + 59T^{2} \) |
| 61 | \( 1 - 10.3T + 61T^{2} \) |
| 67 | \( 1 + 6.36T + 67T^{2} \) |
| 71 | \( 1 + 8.81T + 71T^{2} \) |
| 73 | \( 1 + 2.13T + 73T^{2} \) |
| 79 | \( 1 + 11.9T + 79T^{2} \) |
| 83 | \( 1 + 10.2T + 83T^{2} \) |
| 89 | \( 1 - 6.30T + 89T^{2} \) |
| 97 | \( 1 - 11.9T + 97T^{2} \) |
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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
−8.057624154207794673698158516190, −7.64468671924943935239480745208, −6.40751920912108060831652779027, −5.59281330907452283559610327284, −5.40830051395393997615476780486, −4.36502398343519161848689246449, −3.94605461465103993261280386721, −2.86595066872420315489560880457, −1.61896843403923117030432393856, −1.07357639585980724859272584588,
1.07357639585980724859272584588, 1.61896843403923117030432393856, 2.86595066872420315489560880457, 3.94605461465103993261280386721, 4.36502398343519161848689246449, 5.40830051395393997615476780486, 5.59281330907452283559610327284, 6.40751920912108060831652779027, 7.64468671924943935239480745208, 8.057624154207794673698158516190
