| L(s) = 1 | − 16.2·2-s + 76.4·3-s + 134.·4-s + 312.·5-s − 1.23e3·6-s − 343·7-s − 107.·8-s + 3.65e3·9-s − 5.05e3·10-s − 283.·11-s + 1.02e4·12-s − 1.17e4·13-s + 5.55e3·14-s + 2.38e4·15-s − 1.54e4·16-s + 4.02e3·17-s − 5.91e4·18-s + 3.78e3·19-s + 4.20e4·20-s − 2.62e4·21-s + 4.59e3·22-s − 2.58e3·23-s − 8.19e3·24-s + 1.92e4·25-s + 1.89e5·26-s + 1.11e5·27-s − 4.61e4·28-s + ⋯ |
| L(s) = 1 | − 1.43·2-s + 1.63·3-s + 1.05·4-s + 1.11·5-s − 2.34·6-s − 0.377·7-s − 0.0740·8-s + 1.66·9-s − 1.59·10-s − 0.0642·11-s + 1.71·12-s − 1.47·13-s + 0.541·14-s + 1.82·15-s − 0.945·16-s + 0.198·17-s − 2.39·18-s + 0.126·19-s + 1.17·20-s − 0.617·21-s + 0.0920·22-s − 0.0442·23-s − 0.120·24-s + 0.246·25-s + 2.11·26-s + 1.09·27-s − 0.397·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.116154072\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.116154072\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + 343T \) |
| good | 2 | \( 1 + 16.2T + 128T^{2} \) |
| 3 | \( 1 - 76.4T + 2.18e3T^{2} \) |
| 5 | \( 1 - 312.T + 7.81e4T^{2} \) |
| 11 | \( 1 + 283.T + 1.94e7T^{2} \) |
| 13 | \( 1 + 1.17e4T + 6.27e7T^{2} \) |
| 17 | \( 1 - 4.02e3T + 4.10e8T^{2} \) |
| 19 | \( 1 - 3.78e3T + 8.93e8T^{2} \) |
| 23 | \( 1 + 2.58e3T + 3.40e9T^{2} \) |
| 29 | \( 1 + 1.27e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + 1.59e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 5.83e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 1.85e4T + 1.94e11T^{2} \) |
| 43 | \( 1 + 3.23e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 1.92e3T + 5.06e11T^{2} \) |
| 53 | \( 1 - 1.35e6T + 1.17e12T^{2} \) |
| 59 | \( 1 - 8.68e5T + 2.48e12T^{2} \) |
| 61 | \( 1 + 1.27e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 5.84e5T + 6.06e12T^{2} \) |
| 71 | \( 1 + 5.67e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 4.97e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 8.70e5T + 1.92e13T^{2} \) |
| 83 | \( 1 - 3.87e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 1.02e7T + 4.42e13T^{2} \) |
| 97 | \( 1 - 3.98e6T + 8.07e13T^{2} \) |
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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
−20.33405820931761331150310396742, −19.42753266438982118507604672467, −18.21065847111089065376569316290, −16.73539889738544817238284461214, −14.75283466028508829624793004813, −13.31947393752397283791288230075, −9.938860129374097454142208376625, −9.216437502843415366742090070228, −7.54844764954608328007123844444, −2.21275473409769419015641267435,
2.21275473409769419015641267435, 7.54844764954608328007123844444, 9.216437502843415366742090070228, 9.938860129374097454142208376625, 13.31947393752397283791288230075, 14.75283466028508829624793004813, 16.73539889738544817238284461214, 18.21065847111089065376569316290, 19.42753266438982118507604672467, 20.33405820931761331150310396742
