| L(s) = 1 | + 22·2-s + 21·3-s + 356·4-s + 140·5-s + 462·6-s − 343·7-s + 5.01e3·8-s − 1.74e3·9-s + 3.08e3·10-s + 5.05e3·11-s + 7.47e3·12-s − 2.19e3·13-s − 7.54e3·14-s + 2.94e3·15-s + 6.47e4·16-s + 2.73e4·17-s − 3.84e4·18-s − 3.26e4·19-s + 4.98e4·20-s − 7.20e3·21-s + 1.11e5·22-s − 2.30e4·23-s + 1.05e5·24-s − 5.85e4·25-s − 4.83e4·26-s − 8.25e4·27-s − 1.22e5·28-s + ⋯ |
| L(s) = 1 | + 1.94·2-s + 0.449·3-s + 2.78·4-s + 0.500·5-s + 0.873·6-s − 0.377·7-s + 3.46·8-s − 0.798·9-s + 0.973·10-s + 1.14·11-s + 1.24·12-s − 0.277·13-s − 0.734·14-s + 0.224·15-s + 3.95·16-s + 1.35·17-s − 1.55·18-s − 1.09·19-s + 1.39·20-s − 0.169·21-s + 2.22·22-s − 0.395·23-s + 1.55·24-s − 0.749·25-s − 0.539·26-s − 0.807·27-s − 1.05·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(8.201964588\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.201964588\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + p^{3} T \) |
| 13 | \( 1 + p^{3} T \) |
| good | 2 | \( 1 - 11 p T + p^{7} T^{2} \) |
| 3 | \( 1 - 7 p T + p^{7} T^{2} \) |
| 5 | \( 1 - 28 p T + p^{7} T^{2} \) |
| 11 | \( 1 - 5051 T + p^{7} T^{2} \) |
| 17 | \( 1 - 27384 T + p^{7} T^{2} \) |
| 19 | \( 1 + 32690 T + p^{7} T^{2} \) |
| 23 | \( 1 + 23085 T + p^{7} T^{2} \) |
| 29 | \( 1 + 14068 T + p^{7} T^{2} \) |
| 31 | \( 1 + 203007 T + p^{7} T^{2} \) |
| 37 | \( 1 - 544041 T + p^{7} T^{2} \) |
| 41 | \( 1 + 352079 T + p^{7} T^{2} \) |
| 43 | \( 1 - 340412 T + p^{7} T^{2} \) |
| 47 | \( 1 - 406329 T + p^{7} T^{2} \) |
| 53 | \( 1 + 1909680 T + p^{7} T^{2} \) |
| 59 | \( 1 + 2867214 T + p^{7} T^{2} \) |
| 61 | \( 1 + 216419 T + p^{7} T^{2} \) |
| 67 | \( 1 - 2538043 T + p^{7} T^{2} \) |
| 71 | \( 1 + 2071872 T + p^{7} T^{2} \) |
| 73 | \( 1 - 185913 T + p^{7} T^{2} \) |
| 79 | \( 1 + 954631 T + p^{7} T^{2} \) |
| 83 | \( 1 - 5649672 T + p^{7} T^{2} \) |
| 89 | \( 1 + 4673830 T + p^{7} T^{2} \) |
| 97 | \( 1 + 13686645 T + p^{7} T^{2} \) |
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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
−12.82335145976683507649204402472, −12.01091830576325984977196578932, −10.98440174692348230790418040057, −9.556317392599175389741923589716, −7.76552647548874126676124409088, −6.35709795650353071995654041801, −5.65336986020065899529988686625, −4.11416711914782565940976020985, −3.07964288778612313257754580291, −1.83945285411121143181725198345,
1.83945285411121143181725198345, 3.07964288778612313257754580291, 4.11416711914782565940976020985, 5.65336986020065899529988686625, 6.35709795650353071995654041801, 7.76552647548874126676124409088, 9.556317392599175389741923589716, 10.98440174692348230790418040057, 12.01091830576325984977196578932, 12.82335145976683507649204402472
