| L(s) = 1 | + 3.73·2-s + 5.95·4-s + 3.90·5-s + 36.4·7-s − 7.64·8-s + 14.5·10-s − 19.1·11-s + 13·13-s + 136.·14-s − 76.1·16-s + 83.8·17-s + 46.8·19-s + 23.2·20-s − 71.6·22-s − 103.·23-s − 109.·25-s + 48.5·26-s + 216.·28-s − 108.·29-s − 147.·31-s − 223.·32-s + 313.·34-s + 142.·35-s − 160.·37-s + 175.·38-s − 29.8·40-s − 231.·41-s + ⋯ |
| L(s) = 1 | + 1.32·2-s + 0.744·4-s + 0.349·5-s + 1.96·7-s − 0.337·8-s + 0.461·10-s − 0.526·11-s + 0.277·13-s + 2.59·14-s − 1.19·16-s + 1.19·17-s + 0.565·19-s + 0.260·20-s − 0.694·22-s − 0.941·23-s − 0.877·25-s + 0.366·26-s + 1.46·28-s − 0.693·29-s − 0.854·31-s − 1.23·32-s + 1.58·34-s + 0.687·35-s − 0.710·37-s + 0.747·38-s − 0.118·40-s − 0.881·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.463699110\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.463699110\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 13 | \( 1 - 13T \) |
| good | 2 | \( 1 - 3.73T + 8T^{2} \) |
| 5 | \( 1 - 3.90T + 125T^{2} \) |
| 7 | \( 1 - 36.4T + 343T^{2} \) |
| 11 | \( 1 + 19.1T + 1.33e3T^{2} \) |
| 17 | \( 1 - 83.8T + 4.91e3T^{2} \) |
| 19 | \( 1 - 46.8T + 6.85e3T^{2} \) |
| 23 | \( 1 + 103.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 108.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 147.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 160.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 231.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 340.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 119.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 732.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 229.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 108.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 10.3T + 3.00e5T^{2} \) |
| 71 | \( 1 - 869.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.09e3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 140.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 159.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.06e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 858.T + 9.12e5T^{2} \) |
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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
−13.31021902225716794687207751899, −12.00912012430591650592146708850, −11.44369951122171870161496742257, −10.14101031570273010591850106839, −8.543202226591403593554432334393, −7.48512520755494709512649291224, −5.66736177146343768716477950744, −5.10031613305038362926552409532, −3.74683266888197716678725204756, −1.90815216868512978279357706006,
1.90815216868512978279357706006, 3.74683266888197716678725204756, 5.10031613305038362926552409532, 5.66736177146343768716477950744, 7.48512520755494709512649291224, 8.543202226591403593554432334393, 10.14101031570273010591850106839, 11.44369951122171870161496742257, 12.00912012430591650592146708850, 13.31021902225716794687207751899
