| L(s) = 1 | + 28.6·2-s + 729·3-s − 7.37e3·4-s − 9.30e3·5-s + 2.08e4·6-s + 2.16e5·7-s − 4.46e5·8-s + 5.31e5·9-s − 2.66e5·10-s + 5.39e6·11-s − 5.37e6·12-s + 2.36e7·13-s + 6.20e6·14-s − 6.78e6·15-s + 4.75e7·16-s + 9.98e7·17-s + 1.52e7·18-s + 2.92e8·19-s + 6.86e7·20-s + 1.57e8·21-s + 1.54e8·22-s − 6.28e8·23-s − 3.25e8·24-s − 1.13e9·25-s + 6.79e8·26-s + 3.87e8·27-s − 1.59e9·28-s + ⋯ |
| L(s) = 1 | + 0.316·2-s + 0.577·3-s − 0.899·4-s − 0.266·5-s + 0.182·6-s + 0.695·7-s − 0.601·8-s + 0.333·9-s − 0.0843·10-s + 0.918·11-s − 0.519·12-s + 1.36·13-s + 0.220·14-s − 0.153·15-s + 0.709·16-s + 1.00·17-s + 0.105·18-s + 1.42·19-s + 0.239·20-s + 0.401·21-s + 0.291·22-s − 0.884·23-s − 0.347·24-s − 0.928·25-s + 0.431·26-s + 0.192·27-s − 0.625·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(7)\) |
\(\approx\) |
\(3.492872063\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.492872063\) |
| \(L(\frac{15}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 729T \) |
| 59 | \( 1 + 4.21e10T \) |
| good | 2 | \( 1 - 28.6T + 8.19e3T^{2} \) |
| 5 | \( 1 + 9.30e3T + 1.22e9T^{2} \) |
| 7 | \( 1 - 2.16e5T + 9.68e10T^{2} \) |
| 11 | \( 1 - 5.39e6T + 3.45e13T^{2} \) |
| 13 | \( 1 - 2.36e7T + 3.02e14T^{2} \) |
| 17 | \( 1 - 9.98e7T + 9.90e15T^{2} \) |
| 19 | \( 1 - 2.92e8T + 4.20e16T^{2} \) |
| 23 | \( 1 + 6.28e8T + 5.04e17T^{2} \) |
| 29 | \( 1 + 1.36e9T + 1.02e19T^{2} \) |
| 31 | \( 1 - 3.47e8T + 2.44e19T^{2} \) |
| 37 | \( 1 + 2.74e9T + 2.43e20T^{2} \) |
| 41 | \( 1 + 3.57e10T + 9.25e20T^{2} \) |
| 43 | \( 1 + 4.48e10T + 1.71e21T^{2} \) |
| 47 | \( 1 - 1.39e11T + 5.46e21T^{2} \) |
| 53 | \( 1 + 5.29e9T + 2.60e22T^{2} \) |
| 61 | \( 1 - 3.45e11T + 1.61e23T^{2} \) |
| 67 | \( 1 + 5.99e11T + 5.48e23T^{2} \) |
| 71 | \( 1 + 6.16e11T + 1.16e24T^{2} \) |
| 73 | \( 1 - 1.52e12T + 1.67e24T^{2} \) |
| 79 | \( 1 - 2.64e12T + 4.66e24T^{2} \) |
| 83 | \( 1 - 3.15e12T + 8.87e24T^{2} \) |
| 89 | \( 1 - 5.43e12T + 2.19e25T^{2} \) |
| 97 | \( 1 + 2.80e12T + 6.73e25T^{2} \) |
| 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
−10.13000029121582776070123567130, −9.231977597591575057697068197112, −8.371821056088060397120347058803, −7.62437963611071832717149608771, −6.08537577368903751776773102299, −5.08484398190044031312900029237, −3.83736899365247573146194789238, −3.45938503991247173468472917444, −1.65837915097818559373005534269, −0.802295579230023349482423571110,
0.802295579230023349482423571110, 1.65837915097818559373005534269, 3.45938503991247173468472917444, 3.83736899365247573146194789238, 5.08484398190044031312900029237, 6.08537577368903751776773102299, 7.62437963611071832717149608771, 8.371821056088060397120347058803, 9.231977597591575057697068197112, 10.13000029121582776070123567130
