| L(s) = 1 | + 0.149·2-s − 127.·4-s − 125·5-s + 1.15e3·7-s − 38.3·8-s − 18.7·10-s − 3.97e3·11-s + 1.39e4·13-s + 173.·14-s + 1.63e4·16-s − 5.24e3·17-s + 2.08e4·19-s + 1.59e4·20-s − 596.·22-s − 7.86e4·23-s + 1.56e4·25-s + 2.09e3·26-s − 1.47e5·28-s + 4.07e4·29-s − 4.90e4·31-s + 7.37e3·32-s − 786.·34-s − 1.44e5·35-s + 2.58e5·37-s + 3.13e3·38-s + 4.79e3·40-s − 1.20e5·41-s + ⋯ |
| L(s) = 1 | + 0.0132·2-s − 0.999·4-s − 0.447·5-s + 1.27·7-s − 0.0265·8-s − 0.00592·10-s − 0.900·11-s + 1.76·13-s + 0.0168·14-s + 0.999·16-s − 0.258·17-s + 0.698·19-s + 0.447·20-s − 0.0119·22-s − 1.34·23-s + 0.199·25-s + 0.0233·26-s − 1.27·28-s + 0.310·29-s − 0.295·31-s + 0.0397·32-s − 0.00343·34-s − 0.568·35-s + 0.837·37-s + 0.00926·38-s + 0.0118·40-s − 0.273·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.709579210\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.709579210\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + 125T \) |
| good | 2 | \( 1 - 0.149T + 128T^{2} \) |
| 7 | \( 1 - 1.15e3T + 8.23e5T^{2} \) |
| 11 | \( 1 + 3.97e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 1.39e4T + 6.27e7T^{2} \) |
| 17 | \( 1 + 5.24e3T + 4.10e8T^{2} \) |
| 19 | \( 1 - 2.08e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 7.86e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 4.07e4T + 1.72e10T^{2} \) |
| 31 | \( 1 + 4.90e4T + 2.75e10T^{2} \) |
| 37 | \( 1 - 2.58e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 1.20e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 4.59e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 1.31e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 6.05e5T + 1.17e12T^{2} \) |
| 59 | \( 1 + 2.73e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 1.60e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 2.45e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 4.66e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 2.84e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 4.00e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 9.87e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 1.81e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 1.03e7T + 8.07e13T^{2} \) |
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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
−10.14058286953973582128235683027, −8.947969935356272355312955458617, −8.176407841522933442731081024407, −7.75771960344806485912226629937, −6.10018371163744576010184122357, −5.13711901424858664047483993229, −4.30859697220222466135130790650, −3.36224699513768516899937647483, −1.70154163278061050988988057062, −0.63269227660891068988796224558,
0.63269227660891068988796224558, 1.70154163278061050988988057062, 3.36224699513768516899937647483, 4.30859697220222466135130790650, 5.13711901424858664047483993229, 6.10018371163744576010184122357, 7.75771960344806485912226629937, 8.176407841522933442731081024407, 8.947969935356272355312955458617, 10.14058286953973582128235683027
