| L(s) = 1 | − 0.267·2-s − 3-s − 1.92·4-s + 0.267·6-s + 0.207·7-s + 1.04·8-s + 9-s − 1.27·11-s + 1.92·12-s + 5.37·13-s − 0.0553·14-s + 3.57·16-s − 0.969·17-s − 0.267·18-s + 6.42·19-s − 0.207·21-s + 0.340·22-s − 5.87·23-s − 1.04·24-s − 1.43·26-s − 27-s − 0.399·28-s − 9.93·29-s + 5.97·31-s − 3.05·32-s + 1.27·33-s + 0.259·34-s + ⋯ |
| L(s) = 1 | − 0.188·2-s − 0.577·3-s − 0.964·4-s + 0.109·6-s + 0.0783·7-s + 0.371·8-s + 0.333·9-s − 0.383·11-s + 0.556·12-s + 1.49·13-s − 0.0147·14-s + 0.894·16-s − 0.235·17-s − 0.0629·18-s + 1.47·19-s − 0.0452·21-s + 0.0725·22-s − 1.22·23-s − 0.214·24-s − 0.281·26-s − 0.192·27-s − 0.0755·28-s − 1.84·29-s + 1.07·31-s − 0.540·32-s + 0.221·33-s + 0.0444·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.024806062\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.024806062\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 47 | \( 1 - T \) |
| good | 2 | \( 1 + 0.267T + 2T^{2} \) |
| 7 | \( 1 - 0.207T + 7T^{2} \) |
| 11 | \( 1 + 1.27T + 11T^{2} \) |
| 13 | \( 1 - 5.37T + 13T^{2} \) |
| 17 | \( 1 + 0.969T + 17T^{2} \) |
| 19 | \( 1 - 6.42T + 19T^{2} \) |
| 23 | \( 1 + 5.87T + 23T^{2} \) |
| 29 | \( 1 + 9.93T + 29T^{2} \) |
| 31 | \( 1 - 5.97T + 31T^{2} \) |
| 37 | \( 1 - 1.19T + 37T^{2} \) |
| 41 | \( 1 - 2.24T + 41T^{2} \) |
| 43 | \( 1 - 6.20T + 43T^{2} \) |
| 53 | \( 1 - 8.64T + 53T^{2} \) |
| 59 | \( 1 + 9.70T + 59T^{2} \) |
| 61 | \( 1 + 10.1T + 61T^{2} \) |
| 67 | \( 1 + 4.36T + 67T^{2} \) |
| 71 | \( 1 - 9.54T + 71T^{2} \) |
| 73 | \( 1 - 11.7T + 73T^{2} \) |
| 79 | \( 1 + 15.7T + 79T^{2} \) |
| 83 | \( 1 + 1.79T + 83T^{2} \) |
| 89 | \( 1 + 7.28T + 89T^{2} \) |
| 97 | \( 1 - 0.290T + 97T^{2} \) |
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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
−8.553036737516490207407932762422, −7.895496322336472721776251450537, −7.25539080592416205237553956847, −5.96673716698341771912699336611, −5.75229077171789423552272465412, −4.74472427067016655893150762445, −4.01553204214416328496478119368, −3.24018962172131346501451702935, −1.68948806147203965253899989810, −0.66734418222847149311091553215,
0.66734418222847149311091553215, 1.68948806147203965253899989810, 3.24018962172131346501451702935, 4.01553204214416328496478119368, 4.74472427067016655893150762445, 5.75229077171789423552272465412, 5.96673716698341771912699336611, 7.25539080592416205237553956847, 7.895496322336472721776251450537, 8.553036737516490207407932762422
