| L(s) = 1 | + 8·2-s + 64·4-s + 11.2·5-s + 1.43e3·7-s + 512·8-s + 89.9·10-s + 4.43e3·11-s − 7.23e3·13-s + 1.15e4·14-s + 4.09e3·16-s + 1.32e4·17-s + 1.35e4·19-s + 719.·20-s + 3.54e4·22-s − 6.57e4·23-s − 7.79e4·25-s − 5.78e4·26-s + 9.21e4·28-s + 1.48e5·29-s + 1.38e5·31-s + 3.27e4·32-s + 1.05e5·34-s + 1.61e4·35-s − 1.21e5·37-s + 1.08e5·38-s + 5.75e3·40-s + 5.27e4·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s + 0.0402·5-s + 1.58·7-s + 0.353·8-s + 0.0284·10-s + 1.00·11-s − 0.912·13-s + 1.12·14-s + 0.250·16-s + 0.652·17-s + 0.451·19-s + 0.0201·20-s + 0.710·22-s − 1.12·23-s − 0.998·25-s − 0.645·26-s + 0.793·28-s + 1.13·29-s + 0.837·31-s + 0.176·32-s + 0.461·34-s + 0.0638·35-s − 0.394·37-s + 0.319·38-s + 0.0142·40-s + 0.119·41-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
| \(L(4)\) |
\(\approx\) |
\(4.345515741\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.345515741\) |
| \(L(\frac{9}{2})\) |
|
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 | \( 1 - 11.2T + 7.81e4T^{2} \) |
| 7 | \( 1 - 1.43e3T + 8.23e5T^{2} \) |
| 11 | \( 1 - 4.43e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 7.23e3T + 6.27e7T^{2} \) |
| 17 | \( 1 - 1.32e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 1.35e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 6.57e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 1.48e5T + 1.72e10T^{2} \) |
| 31 | \( 1 - 1.38e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 1.21e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 5.27e4T + 1.94e11T^{2} \) |
| 43 | \( 1 - 7.51e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 9.83e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 6.21e5T + 1.17e12T^{2} \) |
| 59 | \( 1 - 8.14e5T + 2.48e12T^{2} \) |
| 61 | \( 1 + 2.46e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 3.45e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 3.64e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 5.55e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 2.04e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 1.02e7T + 2.71e13T^{2} \) |
| 89 | \( 1 - 8.12e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 1.32e7T + 8.07e13T^{2} \) |
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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
−11.97242999415774659728891233848, −10.76898585998756442859895585286, −9.629562599532676941179174893086, −8.210410971325583294273430752052, −7.38993676371899369711766548090, −6.00314592348007572420920317240, −4.90441507474721044289019496111, −3.98999996144438392466075635809, −2.32767467406576971262759678772, −1.16447941376005439179190034836,
1.16447941376005439179190034836, 2.32767467406576971262759678772, 3.98999996144438392466075635809, 4.90441507474721044289019496111, 6.00314592348007572420920317240, 7.38993676371899369711766548090, 8.210410971325583294273430752052, 9.629562599532676941179174893086, 10.76898585998756442859895585286, 11.97242999415774659728891233848
