| L(s) = 1 | + 8·2-s + 64·4-s + 465.·5-s − 49.7·7-s + 512·8-s + 3.72e3·10-s − 302.·11-s − 5.51e3·13-s − 397.·14-s + 4.09e3·16-s + 2.26e4·17-s + 5.28e4·19-s + 2.98e4·20-s − 2.42e3·22-s − 2.00e4·23-s + 1.38e5·25-s − 4.41e4·26-s − 3.18e3·28-s − 2.50e5·29-s + 2.62e5·31-s + 3.27e4·32-s + 1.81e5·34-s − 2.31e4·35-s − 4.60e5·37-s + 4.22e5·38-s + 2.38e5·40-s + 2.40e5·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s + 1.66·5-s − 0.0548·7-s + 0.353·8-s + 1.17·10-s − 0.0686·11-s − 0.696·13-s − 0.0387·14-s + 0.250·16-s + 1.11·17-s + 1.76·19-s + 0.833·20-s − 0.0485·22-s − 0.343·23-s + 1.77·25-s − 0.492·26-s − 0.0274·28-s − 1.90·29-s + 1.58·31-s + 0.176·32-s + 0.790·34-s − 0.0913·35-s − 1.49·37-s + 1.24·38-s + 0.589·40-s + 0.544·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(3.764188434\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.764188434\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 8T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 465.T + 7.81e4T^{2} \) |
| 7 | \( 1 + 49.7T + 8.23e5T^{2} \) |
| 11 | \( 1 + 302.T + 1.94e7T^{2} \) |
| 13 | \( 1 + 5.51e3T + 6.27e7T^{2} \) |
| 17 | \( 1 - 2.26e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 5.28e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 2.00e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 2.50e5T + 1.72e10T^{2} \) |
| 31 | \( 1 - 2.62e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 4.60e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 2.40e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 5.68e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 8.75e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 8.11e5T + 1.17e12T^{2} \) |
| 59 | \( 1 - 1.29e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 1.62e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 1.84e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 4.56e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 1.91e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 1.81e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 3.08e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 4.10e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 7.91e6T + 8.07e13T^{2} \) |
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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
−13.90950599947658951664704073546, −12.95786390292402414818385045647, −11.79280731637399444260730420087, −10.20831092861733402294486381292, −9.461337073081180425858300970244, −7.46520902792111729748488978177, −5.98212604342142745302997430887, −5.12905809123235708668941095833, −3.02168254800358012955589067168, −1.55726676827836595875658551509,
1.55726676827836595875658551509, 3.02168254800358012955589067168, 5.12905809123235708668941095833, 5.98212604342142745302997430887, 7.46520902792111729748488978177, 9.461337073081180425858300970244, 10.20831092861733402294486381292, 11.79280731637399444260730420087, 12.95786390292402414818385045647, 13.90950599947658951664704073546
