| L(s) = 1 | − 64·2-s + 1.10e3·3-s + 4.09e3·4-s + 5.58e4·5-s − 7.09e4·6-s + 1.17e5·7-s − 2.62e5·8-s − 3.64e5·9-s − 3.57e6·10-s + 6.34e6·11-s + 4.54e6·12-s − 3.04e7·13-s − 7.52e6·14-s + 6.19e7·15-s + 1.67e7·16-s + 1.90e8·17-s + 2.33e7·18-s + 2.23e8·19-s + 2.28e8·20-s + 1.30e8·21-s − 4.05e8·22-s − 2.75e8·23-s − 2.90e8·24-s + 1.90e9·25-s + 1.94e9·26-s − 2.17e9·27-s + 4.81e8·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.878·3-s + 0.5·4-s + 1.59·5-s − 0.620·6-s + 0.377·7-s − 0.353·8-s − 0.228·9-s − 1.13·10-s + 1.07·11-s + 0.439·12-s − 1.74·13-s − 0.267·14-s + 1.40·15-s + 0.250·16-s + 1.90·17-s + 0.161·18-s + 1.08·19-s + 0.799·20-s + 0.331·21-s − 0.763·22-s − 0.387·23-s − 0.310·24-s + 1.55·25-s + 1.23·26-s − 1.07·27-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
| \(L(7)\) |
\(\approx\) |
\(2.391591816\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.391591816\) |
| \(L(\frac{15}{2})\) |
|
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 | \( 1 - 1.10e3T + 1.59e6T^{2} \) |
| 5 | \( 1 - 5.58e4T + 1.22e9T^{2} \) |
| 11 | \( 1 - 6.34e6T + 3.45e13T^{2} \) |
| 13 | \( 1 + 3.04e7T + 3.02e14T^{2} \) |
| 17 | \( 1 - 1.90e8T + 9.90e15T^{2} \) |
| 19 | \( 1 - 2.23e8T + 4.20e16T^{2} \) |
| 23 | \( 1 + 2.75e8T + 5.04e17T^{2} \) |
| 29 | \( 1 + 1.75e9T + 1.02e19T^{2} \) |
| 31 | \( 1 - 3.31e9T + 2.44e19T^{2} \) |
| 37 | \( 1 + 6.03e9T + 2.43e20T^{2} \) |
| 41 | \( 1 - 3.70e10T + 9.25e20T^{2} \) |
| 43 | \( 1 + 9.97e9T + 1.71e21T^{2} \) |
| 47 | \( 1 - 9.22e10T + 5.46e21T^{2} \) |
| 53 | \( 1 + 1.82e11T + 2.60e22T^{2} \) |
| 59 | \( 1 + 9.53e10T + 1.04e23T^{2} \) |
| 61 | \( 1 + 1.55e11T + 1.61e23T^{2} \) |
| 67 | \( 1 + 3.27e11T + 5.48e23T^{2} \) |
| 71 | \( 1 + 1.50e12T + 1.16e24T^{2} \) |
| 73 | \( 1 - 6.71e11T + 1.67e24T^{2} \) |
| 79 | \( 1 + 2.82e12T + 4.66e24T^{2} \) |
| 83 | \( 1 + 1.88e12T + 8.87e24T^{2} \) |
| 89 | \( 1 + 7.04e9T + 2.19e25T^{2} \) |
| 97 | \( 1 - 1.02e12T + 6.73e25T^{2} \) |
| 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
−16.85919165503121449170139691220, −14.56420954578669134812072842553, −14.08468152455767563055538248503, −12.07090609018367119565433109264, −9.927883485928901000490812533613, −9.273817844851678909029607325188, −7.57375133227925065564901946608, −5.64920381359401383660278200266, −2.76894724655451530084241449582, −1.44925018076931525744061969641,
1.44925018076931525744061969641, 2.76894724655451530084241449582, 5.64920381359401383660278200266, 7.57375133227925065564901946608, 9.273817844851678909029607325188, 9.927883485928901000490812533613, 12.07090609018367119565433109264, 14.08468152455767563055538248503, 14.56420954578669134812072842553, 16.85919165503121449170139691220
