| L(s) = 1 | − 14.5·2-s + 84.9·4-s + 125·5-s − 343·7-s + 628.·8-s − 1.82e3·10-s + 8.20e3·11-s + 2.45e3·13-s + 5.00e3·14-s − 2.00e4·16-s + 2.21e4·17-s + 3.82e4·19-s + 1.06e4·20-s − 1.19e5·22-s + 4.89e4·23-s + 1.56e4·25-s − 3.57e4·26-s − 2.91e4·28-s + 8.81e4·29-s − 2.17e4·31-s + 2.12e5·32-s − 3.22e5·34-s − 4.28e4·35-s + 4.92e5·37-s − 5.58e5·38-s + 7.85e4·40-s + 6.39e5·41-s + ⋯ |
| L(s) = 1 | − 1.28·2-s + 0.663·4-s + 0.447·5-s − 0.377·7-s + 0.433·8-s − 0.576·10-s + 1.85·11-s + 0.309·13-s + 0.487·14-s − 1.22·16-s + 1.09·17-s + 1.27·19-s + 0.296·20-s − 2.39·22-s + 0.838·23-s + 0.199·25-s − 0.399·26-s − 0.250·28-s + 0.670·29-s − 0.130·31-s + 1.14·32-s − 1.40·34-s − 0.169·35-s + 1.59·37-s − 1.64·38-s + 0.194·40-s + 1.44·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.604918870\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.604918870\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - 125T \) |
| 7 | \( 1 + 343T \) |
| good | 2 | \( 1 + 14.5T + 128T^{2} \) |
| 11 | \( 1 - 8.20e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 2.45e3T + 6.27e7T^{2} \) |
| 17 | \( 1 - 2.21e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 3.82e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 4.89e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 8.81e4T + 1.72e10T^{2} \) |
| 31 | \( 1 + 2.17e4T + 2.75e10T^{2} \) |
| 37 | \( 1 - 4.92e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 6.39e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 1.91e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 1.63e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 8.30e5T + 1.17e12T^{2} \) |
| 59 | \( 1 - 9.95e5T + 2.48e12T^{2} \) |
| 61 | \( 1 - 1.02e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 4.40e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 4.62e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 1.34e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 3.62e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 1.63e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 3.83e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 2.79e6T + 8.07e13T^{2} \) |
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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
−10.05230440801939721934502885358, −9.447438343560181917560537559812, −8.880898403972746856806330637257, −7.71441876285169979117678183471, −6.82871151696415169390679356341, −5.83420352007061922317428107096, −4.34828524032710868204742891102, −3.03969365025535124716053328708, −1.38606627780954706214194499993, −0.891921022649445418425196044003,
0.891921022649445418425196044003, 1.38606627780954706214194499993, 3.03969365025535124716053328708, 4.34828524032710868204742891102, 5.83420352007061922317428107096, 6.82871151696415169390679356341, 7.71441876285169979117678183471, 8.880898403972746856806330637257, 9.447438343560181917560537559812, 10.05230440801939721934502885358
