| L(s) = 1 | + 90.0·2-s + 243·3-s + 6.05e3·4-s − 4.74e3·5-s + 2.18e4·6-s + 2.12e4·7-s + 3.60e5·8-s + 5.90e4·9-s − 4.27e5·10-s − 5.88e5·11-s + 1.47e6·12-s + 6.21e5·13-s + 1.91e6·14-s − 1.15e6·15-s + 2.00e7·16-s + 3.27e6·17-s + 5.31e6·18-s + 1.04e7·19-s − 2.87e7·20-s + 5.16e6·21-s − 5.29e7·22-s − 1.01e7·23-s + 8.76e7·24-s − 2.63e7·25-s + 5.59e7·26-s + 1.43e7·27-s + 1.28e8·28-s + ⋯ |
| L(s) = 1 | + 1.98·2-s + 0.577·3-s + 2.95·4-s − 0.678·5-s + 1.14·6-s + 0.478·7-s + 3.89·8-s + 0.333·9-s − 1.35·10-s − 1.10·11-s + 1.70·12-s + 0.464·13-s + 0.951·14-s − 0.391·15-s + 4.78·16-s + 0.560·17-s + 0.663·18-s + 0.969·19-s − 2.00·20-s + 0.276·21-s − 2.19·22-s − 0.328·23-s + 2.24·24-s − 0.539·25-s + 0.923·26-s + 0.192·27-s + 1.41·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(11.81459282\) |
| \(L(\frac12)\) |
\(\approx\) |
\(11.81459282\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 243T \) |
| 59 | \( 1 - 7.14e8T \) |
| good | 2 | \( 1 - 90.0T + 2.04e3T^{2} \) |
| 5 | \( 1 + 4.74e3T + 4.88e7T^{2} \) |
| 7 | \( 1 - 2.12e4T + 1.97e9T^{2} \) |
| 11 | \( 1 + 5.88e5T + 2.85e11T^{2} \) |
| 13 | \( 1 - 6.21e5T + 1.79e12T^{2} \) |
| 17 | \( 1 - 3.27e6T + 3.42e13T^{2} \) |
| 19 | \( 1 - 1.04e7T + 1.16e14T^{2} \) |
| 23 | \( 1 + 1.01e7T + 9.52e14T^{2} \) |
| 29 | \( 1 - 1.61e8T + 1.22e16T^{2} \) |
| 31 | \( 1 + 2.14e7T + 2.54e16T^{2} \) |
| 37 | \( 1 - 5.53e8T + 1.77e17T^{2} \) |
| 41 | \( 1 - 8.75e8T + 5.50e17T^{2} \) |
| 43 | \( 1 - 1.09e8T + 9.29e17T^{2} \) |
| 47 | \( 1 + 1.69e9T + 2.47e18T^{2} \) |
| 53 | \( 1 + 6.43e8T + 9.26e18T^{2} \) |
| 61 | \( 1 + 4.14e9T + 4.35e19T^{2} \) |
| 67 | \( 1 - 5.36e9T + 1.22e20T^{2} \) |
| 71 | \( 1 + 1.79e10T + 2.31e20T^{2} \) |
| 73 | \( 1 + 9.32e9T + 3.13e20T^{2} \) |
| 79 | \( 1 - 2.29e9T + 7.47e20T^{2} \) |
| 83 | \( 1 - 4.59e10T + 1.28e21T^{2} \) |
| 89 | \( 1 - 3.18e10T + 2.77e21T^{2} \) |
| 97 | \( 1 + 9.75e10T + 7.15e21T^{2} \) |
| 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
−11.12319439281487599551313886602, −10.09470734231381061795310665471, −7.957247992902341271732276108107, −7.62415101488003988471604602677, −6.24279637448218927123373062966, −5.16651065456003227562538133146, −4.30456753768966342933968208704, −3.31271083244166502046013935235, −2.52604353135594794728926606607, −1.24277591746954512852897234463,
1.24277591746954512852897234463, 2.52604353135594794728926606607, 3.31271083244166502046013935235, 4.30456753768966342933968208704, 5.16651065456003227562538133146, 6.24279637448218927123373062966, 7.62415101488003988471604602677, 7.957247992902341271732276108107, 10.09470734231381061795310665471, 11.12319439281487599551313886602
