| L(s) = 1 | − 8·2-s + 64·4-s + 320.·5-s + 1.19e3·7-s − 512·8-s − 2.56e3·10-s + 6.80e3·11-s + 7.47e3·13-s − 9.55e3·14-s + 4.09e3·16-s + 3.45e4·17-s + 1.79e4·19-s + 2.05e4·20-s − 5.44e4·22-s − 1.70e4·23-s + 2.46e4·25-s − 5.97e4·26-s + 7.64e4·28-s − 7.39e4·29-s − 5.16e4·31-s − 3.27e4·32-s − 2.76e5·34-s + 3.82e5·35-s − 5.17e5·37-s − 1.43e5·38-s − 1.64e5·40-s + 2.03e5·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + 1.14·5-s + 1.31·7-s − 0.353·8-s − 0.810·10-s + 1.54·11-s + 0.943·13-s − 0.930·14-s + 0.250·16-s + 1.70·17-s + 0.601·19-s + 0.573·20-s − 1.09·22-s − 0.291·23-s + 0.315·25-s − 0.667·26-s + 0.657·28-s − 0.562·29-s − 0.311·31-s − 0.176·32-s − 1.20·34-s + 1.50·35-s − 1.68·37-s − 0.425·38-s − 0.405·40-s + 0.460·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(2.772062761\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.772062761\) |
| \(L(\frac{9}{2})\) |
|
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 | \( 1 - 320.T + 7.81e4T^{2} \) |
| 7 | \( 1 - 1.19e3T + 8.23e5T^{2} \) |
| 11 | \( 1 - 6.80e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 7.47e3T + 6.27e7T^{2} \) |
| 17 | \( 1 - 3.45e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 1.79e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 1.70e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 7.39e4T + 1.72e10T^{2} \) |
| 31 | \( 1 + 5.16e4T + 2.75e10T^{2} \) |
| 37 | \( 1 + 5.17e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 2.03e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 9.73e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 7.37e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 5.08e5T + 1.17e12T^{2} \) |
| 59 | \( 1 + 1.90e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 2.07e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 2.98e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 2.98e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 5.26e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 1.39e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 5.23e5T + 2.71e13T^{2} \) |
| 89 | \( 1 + 4.57e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 1.02e7T + 8.07e13T^{2} \) |
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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
−11.45570590780023874320557567955, −10.41021592605496545386155357545, −9.464499930887899250102294880192, −8.622916873325483920549799157987, −7.52502321592806223024002675380, −6.22930630098161077520130002615, −5.30663453183460788678526653635, −3.57807503794106469640820012141, −1.69357054407533087992533680894, −1.26078895091058140871697380014,
1.26078895091058140871697380014, 1.69357054407533087992533680894, 3.57807503794106469640820012141, 5.30663453183460788678526653635, 6.22930630098161077520130002615, 7.52502321592806223024002675380, 8.622916873325483920549799157987, 9.464499930887899250102294880192, 10.41021592605496545386155357545, 11.45570590780023874320557567955
