| L(s) = 1 | + 8.50·2-s − 439.·4-s + 625·5-s + 2.40e3·7-s − 8.09e3·8-s + 5.31e3·10-s + 8.47e4·11-s + 1.89e5·13-s + 2.04e4·14-s + 1.56e5·16-s + 2.38e5·17-s + 6.21e5·19-s − 2.74e5·20-s + 7.20e5·22-s + 2.11e5·23-s + 3.90e5·25-s + 1.60e6·26-s − 1.05e6·28-s − 3.28e6·29-s − 7.79e6·31-s + 5.47e6·32-s + 2.03e6·34-s + 1.50e6·35-s − 8.19e6·37-s + 5.28e6·38-s − 5.06e6·40-s − 1.82e7·41-s + ⋯ |
| L(s) = 1 | + 0.376·2-s − 0.858·4-s + 0.447·5-s + 0.377·7-s − 0.698·8-s + 0.168·10-s + 1.74·11-s + 1.83·13-s + 0.142·14-s + 0.595·16-s + 0.693·17-s + 1.09·19-s − 0.383·20-s + 0.655·22-s + 0.157·23-s + 0.200·25-s + 0.690·26-s − 0.324·28-s − 0.863·29-s − 1.51·31-s + 0.922·32-s + 0.260·34-s + 0.169·35-s − 0.719·37-s + 0.411·38-s − 0.312·40-s − 1.00·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(3.531931798\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.531931798\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - 625T \) |
| 7 | \( 1 - 2.40e3T \) |
| good | 2 | \( 1 - 8.50T + 512T^{2} \) |
| 11 | \( 1 - 8.47e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 1.89e5T + 1.06e10T^{2} \) |
| 17 | \( 1 - 2.38e5T + 1.18e11T^{2} \) |
| 19 | \( 1 - 6.21e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 2.11e5T + 1.80e12T^{2} \) |
| 29 | \( 1 + 3.28e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 7.79e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 8.19e6T + 1.29e14T^{2} \) |
| 41 | \( 1 + 1.82e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 8.54e6T + 5.02e14T^{2} \) |
| 47 | \( 1 - 2.44e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 3.12e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 1.53e8T + 8.66e15T^{2} \) |
| 61 | \( 1 - 9.34e7T + 1.16e16T^{2} \) |
| 67 | \( 1 + 2.45e6T + 2.72e16T^{2} \) |
| 71 | \( 1 - 2.39e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 3.22e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 6.41e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 7.07e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 3.67e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 5.71e8T + 7.60e17T^{2} \) |
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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
−9.927883505644012972841207071481, −9.049864813953083764060103262545, −8.554727585174264509291015255019, −7.13270678401720752635588382538, −5.95815463343669347720835906711, −5.32958641690856916378294451496, −3.90931406492034298259261148142, −3.50403049535778258343199644681, −1.57380696013011510169374513453, −0.884134756607568551136197000310,
0.884134756607568551136197000310, 1.57380696013011510169374513453, 3.50403049535778258343199644681, 3.90931406492034298259261148142, 5.32958641690856916378294451496, 5.95815463343669347720835906711, 7.13270678401720752635588382538, 8.554727585174264509291015255019, 9.049864813953083764060103262545, 9.927883505644012972841207071481
