| L(s) = 1 | − 8·2-s + 64·4-s − 325.·5-s − 1.45e3·7-s − 512·8-s + 2.60e3·10-s + 5.55e3·11-s + 1.35e4·13-s + 1.16e4·14-s + 4.09e3·16-s − 1.18e4·17-s + 4.45e4·19-s − 2.08e4·20-s − 4.44e4·22-s − 2.68e4·23-s + 2.78e4·25-s − 1.08e5·26-s − 9.28e4·28-s − 1.40e5·29-s + 1.65e5·31-s − 3.27e4·32-s + 9.46e4·34-s + 4.72e5·35-s + 3.63e4·37-s − 3.56e5·38-s + 1.66e5·40-s + 4.78e5·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s − 1.16·5-s − 1.59·7-s − 0.353·8-s + 0.823·10-s + 1.25·11-s + 1.70·13-s + 1.13·14-s + 0.250·16-s − 0.583·17-s + 1.48·19-s − 0.582·20-s − 0.889·22-s − 0.459·23-s + 0.356·25-s − 1.20·26-s − 0.799·28-s − 1.07·29-s + 0.998·31-s − 0.176·32-s + 0.412·34-s + 1.86·35-s + 0.117·37-s − 1.05·38-s + 0.411·40-s + 1.08·41-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
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
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 | \( 1 + 325.T + 7.81e4T^{2} \) |
| 7 | \( 1 + 1.45e3T + 8.23e5T^{2} \) |
| 11 | \( 1 - 5.55e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 1.35e4T + 6.27e7T^{2} \) |
| 17 | \( 1 + 1.18e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 4.45e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 2.68e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 1.40e5T + 1.72e10T^{2} \) |
| 31 | \( 1 - 1.65e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 3.63e4T + 9.49e10T^{2} \) |
| 41 | \( 1 - 4.78e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 6.04e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 5.03e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 1.99e6T + 1.17e12T^{2} \) |
| 59 | \( 1 + 6.09e5T + 2.48e12T^{2} \) |
| 61 | \( 1 - 1.90e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 1.77e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 1.00e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 1.46e5T + 1.10e13T^{2} \) |
| 79 | \( 1 + 7.27e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 5.37e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 2.94e6T + 4.42e13T^{2} \) |
| 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
−11.20085595331625711486415438289, −9.799840546048895810980123722792, −9.074757220379804907551173097935, −8.020161533218158218577496579493, −6.83532801976724897536650279082, −6.10266754687602312574599926942, −3.91084765377856554516121489971, −3.25821024485899897914048814124, −1.17622267863994156066916097872, 0,
1.17622267863994156066916097872, 3.25821024485899897914048814124, 3.91084765377856554516121489971, 6.10266754687602312574599926942, 6.83532801976724897536650279082, 8.020161533218158218577496579493, 9.074757220379804907551173097935, 9.799840546048895810980123722792, 11.20085595331625711486415438289
