| L(s) = 1 | + 64·2-s − 852.·3-s + 4.09e3·4-s + 2.68e4·5-s − 5.45e4·6-s − 1.17e5·7-s + 2.62e5·8-s − 8.68e5·9-s + 1.71e6·10-s + 1.14e7·11-s − 3.49e6·12-s + 2.15e7·13-s − 7.52e6·14-s − 2.28e7·15-s + 1.67e7·16-s + 1.20e8·17-s − 5.55e7·18-s − 9.08e7·19-s + 1.09e8·20-s + 1.00e8·21-s + 7.29e8·22-s + 1.07e9·23-s − 2.23e8·24-s − 5.02e8·25-s + 1.37e9·26-s + 2.09e9·27-s − 4.81e8·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.674·3-s + 0.5·4-s + 0.767·5-s − 0.477·6-s − 0.377·7-s + 0.353·8-s − 0.544·9-s + 0.542·10-s + 1.94·11-s − 0.337·12-s + 1.23·13-s − 0.267·14-s − 0.517·15-s + 0.250·16-s + 1.20·17-s − 0.384·18-s − 0.442·19-s + 0.383·20-s + 0.255·21-s + 1.37·22-s + 1.51·23-s − 0.238·24-s − 0.411·25-s + 0.875·26-s + 1.04·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(7)\) |
\(\approx\) |
\(2.721953028\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.721953028\) |
| \(L(\frac{15}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 64T \) |
| 7 | \( 1 + 1.17e5T \) |
| good | 3 | \( 1 + 852.T + 1.59e6T^{2} \) |
| 5 | \( 1 - 2.68e4T + 1.22e9T^{2} \) |
| 11 | \( 1 - 1.14e7T + 3.45e13T^{2} \) |
| 13 | \( 1 - 2.15e7T + 3.02e14T^{2} \) |
| 17 | \( 1 - 1.20e8T + 9.90e15T^{2} \) |
| 19 | \( 1 + 9.08e7T + 4.20e16T^{2} \) |
| 23 | \( 1 - 1.07e9T + 5.04e17T^{2} \) |
| 29 | \( 1 + 3.12e9T + 1.02e19T^{2} \) |
| 31 | \( 1 + 2.89e9T + 2.44e19T^{2} \) |
| 37 | \( 1 + 9.56e9T + 2.43e20T^{2} \) |
| 41 | \( 1 + 3.69e9T + 9.25e20T^{2} \) |
| 43 | \( 1 + 4.09e10T + 1.71e21T^{2} \) |
| 47 | \( 1 - 1.30e11T + 5.46e21T^{2} \) |
| 53 | \( 1 + 1.52e11T + 2.60e22T^{2} \) |
| 59 | \( 1 + 8.12e10T + 1.04e23T^{2} \) |
| 61 | \( 1 - 5.04e10T + 1.61e23T^{2} \) |
| 67 | \( 1 + 7.97e11T + 5.48e23T^{2} \) |
| 71 | \( 1 - 7.61e11T + 1.16e24T^{2} \) |
| 73 | \( 1 - 7.61e10T + 1.67e24T^{2} \) |
| 79 | \( 1 + 6.85e11T + 4.66e24T^{2} \) |
| 83 | \( 1 - 5.01e12T + 8.87e24T^{2} \) |
| 89 | \( 1 - 5.72e12T + 2.19e25T^{2} \) |
| 97 | \( 1 + 1.46e13T + 6.73e25T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−16.59066243112664313295992909039, −14.74217677861886266067719379845, −13.62019152375843787741530662030, −12.13869661065416120997435127670, −10.95543501582612527549997294990, −9.143035437524972572979932043919, −6.55890438773098246956921348084, −5.62233172202848850906630404643, −3.57168253420461065984679219945, −1.31090780903897985860164092491,
1.31090780903897985860164092491, 3.57168253420461065984679219945, 5.62233172202848850906630404643, 6.55890438773098246956921348084, 9.143035437524972572979932043919, 10.95543501582612527549997294990, 12.13869661065416120997435127670, 13.62019152375843787741530662030, 14.74217677861886266067719379845, 16.59066243112664313295992909039
