| L(s) = 1 | − 1.85·2-s + 3-s + 1.43·4-s + 5-s − 1.85·6-s + 4.25·7-s + 1.04·8-s + 9-s − 1.85·10-s − 1.39·11-s + 1.43·12-s + 2.97·13-s − 7.88·14-s + 15-s − 4.80·16-s − 5.65·17-s − 1.85·18-s − 7.04·19-s + 1.43·20-s + 4.25·21-s + 2.58·22-s + 1.04·24-s + 25-s − 5.50·26-s + 27-s + 6.11·28-s + 1.32·29-s + ⋯ |
| L(s) = 1 | − 1.31·2-s + 0.577·3-s + 0.719·4-s + 0.447·5-s − 0.756·6-s + 1.60·7-s + 0.368·8-s + 0.333·9-s − 0.586·10-s − 0.420·11-s + 0.415·12-s + 0.824·13-s − 2.10·14-s + 0.258·15-s − 1.20·16-s − 1.37·17-s − 0.437·18-s − 1.61·19-s + 0.321·20-s + 0.927·21-s + 0.550·22-s + 0.212·24-s + 0.200·25-s − 1.08·26-s + 0.192·27-s + 1.15·28-s + 0.246·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7935 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7935 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 23 | \( 1 \) |
| good | 2 | \( 1 + 1.85T + 2T^{2} \) |
| 7 | \( 1 - 4.25T + 7T^{2} \) |
| 11 | \( 1 + 1.39T + 11T^{2} \) |
| 13 | \( 1 - 2.97T + 13T^{2} \) |
| 17 | \( 1 + 5.65T + 17T^{2} \) |
| 19 | \( 1 + 7.04T + 19T^{2} \) |
| 29 | \( 1 - 1.32T + 29T^{2} \) |
| 31 | \( 1 - 3.56T + 31T^{2} \) |
| 37 | \( 1 + 3.67T + 37T^{2} \) |
| 41 | \( 1 + 9.81T + 41T^{2} \) |
| 43 | \( 1 + 10.7T + 43T^{2} \) |
| 47 | \( 1 + 7.10T + 47T^{2} \) |
| 53 | \( 1 + 8.09T + 53T^{2} \) |
| 59 | \( 1 + 0.141T + 59T^{2} \) |
| 61 | \( 1 - 3.62T + 61T^{2} \) |
| 67 | \( 1 - 12.0T + 67T^{2} \) |
| 71 | \( 1 - 10.1T + 71T^{2} \) |
| 73 | \( 1 + 9.15T + 73T^{2} \) |
| 79 | \( 1 + 6.16T + 79T^{2} \) |
| 83 | \( 1 + 16.6T + 83T^{2} \) |
| 89 | \( 1 + 8.30T + 89T^{2} \) |
| 97 | \( 1 - 0.605T + 97T^{2} \) |
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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
−8.064715738964654574837268410717, −6.79128469151074077758389809070, −6.63583995912006891584848419137, −5.27498985626401777562020457955, −4.65739375573816177689553454686, −4.01880126581915723003582534097, −2.66896136259182718035499609001, −1.80221336580352276484681818048, −1.51556163458169570456253192391, 0,
1.51556163458169570456253192391, 1.80221336580352276484681818048, 2.66896136259182718035499609001, 4.01880126581915723003582534097, 4.65739375573816177689553454686, 5.27498985626401777562020457955, 6.63583995912006891584848419137, 6.79128469151074077758389809070, 8.064715738964654574837268410717
