| L(s) = 1 | + 2.37·2-s − 2.37·4-s + 5·5-s + 7.37·7-s − 24.6·8-s + 11.8·10-s − 4.13·11-s − 78.9·13-s + 17.4·14-s − 39.3·16-s − 33.3·17-s + 89.3·19-s − 11.8·20-s − 9.81·22-s − 199.·23-s + 25·25-s − 187.·26-s − 17.4·28-s + 50.4·29-s − 6·31-s + 103.·32-s − 79.0·34-s + 36.8·35-s − 290.·37-s + 211.·38-s − 123.·40-s − 53.3·41-s + ⋯ |
| L(s) = 1 | + 0.838·2-s − 0.296·4-s + 0.447·5-s + 0.398·7-s − 1.08·8-s + 0.375·10-s − 0.113·11-s − 1.68·13-s + 0.333·14-s − 0.615·16-s − 0.475·17-s + 1.07·19-s − 0.132·20-s − 0.0951·22-s − 1.80·23-s + 0.200·25-s − 1.41·26-s − 0.118·28-s + 0.323·29-s − 0.0347·31-s + 0.571·32-s − 0.398·34-s + 0.178·35-s − 1.28·37-s + 0.904·38-s − 0.486·40-s − 0.203·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - 5T \) |
| good | 2 | \( 1 - 2.37T + 8T^{2} \) |
| 7 | \( 1 - 7.37T + 343T^{2} \) |
| 11 | \( 1 + 4.13T + 1.33e3T^{2} \) |
| 13 | \( 1 + 78.9T + 2.19e3T^{2} \) |
| 17 | \( 1 + 33.3T + 4.91e3T^{2} \) |
| 19 | \( 1 - 89.3T + 6.85e3T^{2} \) |
| 23 | \( 1 + 199.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 50.4T + 2.43e4T^{2} \) |
| 31 | \( 1 + 6T + 2.97e4T^{2} \) |
| 37 | \( 1 + 290.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 53.3T + 6.89e4T^{2} \) |
| 43 | \( 1 + 298.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 418.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 399.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 98.2T + 2.05e5T^{2} \) |
| 61 | \( 1 + 683.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 225.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 512.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 994.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 201.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.10e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 372.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 139.T + 9.12e5T^{2} \) |
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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
−10.18968925532739595652587963361, −9.640904232392103980889346750376, −8.560437077153402890418273090342, −7.50137362910994330014738642677, −6.32431035508479658024644789136, −5.23699256741440599580335964172, −4.66977911570974299413027372415, −3.34138618707949579766015942790, −2.06929188837547122907173197286, 0,
2.06929188837547122907173197286, 3.34138618707949579766015942790, 4.66977911570974299413027372415, 5.23699256741440599580335964172, 6.32431035508479658024644789136, 7.50137362910994330014738642677, 8.560437077153402890418273090342, 9.640904232392103980889346750376, 10.18968925532739595652587963361
