| L(s) = 1 | − 60.8·2-s − 243·3-s + 1.65e3·4-s − 4.96e3·5-s + 1.47e4·6-s − 8.70e4·7-s + 2.36e4·8-s + 5.90e4·9-s + 3.02e5·10-s − 6.83e5·11-s − 4.03e5·12-s − 9.74e5·13-s + 5.30e6·14-s + 1.20e6·15-s − 4.83e6·16-s − 6.81e6·17-s − 3.59e6·18-s + 1.81e7·19-s − 8.23e6·20-s + 2.11e7·21-s + 4.15e7·22-s − 3.49e7·23-s − 5.74e6·24-s − 2.42e7·25-s + 5.93e7·26-s − 1.43e7·27-s − 1.44e8·28-s + ⋯ |
| L(s) = 1 | − 1.34·2-s − 0.577·3-s + 0.810·4-s − 0.709·5-s + 0.776·6-s − 1.95·7-s + 0.255·8-s + 0.333·9-s + 0.955·10-s − 1.27·11-s − 0.467·12-s − 0.727·13-s + 2.63·14-s + 0.409·15-s − 1.15·16-s − 1.16·17-s − 0.448·18-s + 1.67·19-s − 0.575·20-s + 1.13·21-s + 1.72·22-s − 1.13·23-s − 0.147·24-s − 0.495·25-s + 0.979·26-s − 0.192·27-s − 1.58·28-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
| \(L(6)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{13}{2})\) |
|
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 | \( 1 + 60.8T + 2.04e3T^{2} \) |
| 5 | \( 1 + 4.96e3T + 4.88e7T^{2} \) |
| 7 | \( 1 + 8.70e4T + 1.97e9T^{2} \) |
| 11 | \( 1 + 6.83e5T + 2.85e11T^{2} \) |
| 13 | \( 1 + 9.74e5T + 1.79e12T^{2} \) |
| 17 | \( 1 + 6.81e6T + 3.42e13T^{2} \) |
| 19 | \( 1 - 1.81e7T + 1.16e14T^{2} \) |
| 23 | \( 1 + 3.49e7T + 9.52e14T^{2} \) |
| 29 | \( 1 - 9.30e7T + 1.22e16T^{2} \) |
| 31 | \( 1 + 1.05e8T + 2.54e16T^{2} \) |
| 37 | \( 1 + 2.32e8T + 1.77e17T^{2} \) |
| 41 | \( 1 - 1.02e9T + 5.50e17T^{2} \) |
| 43 | \( 1 + 1.24e9T + 9.29e17T^{2} \) |
| 47 | \( 1 + 2.18e9T + 2.47e18T^{2} \) |
| 53 | \( 1 + 4.40e9T + 9.26e18T^{2} \) |
| 61 | \( 1 - 8.44e9T + 4.35e19T^{2} \) |
| 67 | \( 1 - 2.45e9T + 1.22e20T^{2} \) |
| 71 | \( 1 - 1.56e10T + 2.31e20T^{2} \) |
| 73 | \( 1 - 1.89e10T + 3.13e20T^{2} \) |
| 79 | \( 1 - 1.99e10T + 7.47e20T^{2} \) |
| 83 | \( 1 - 6.31e10T + 1.28e21T^{2} \) |
| 89 | \( 1 - 1.84e10T + 2.77e21T^{2} \) |
| 97 | \( 1 - 4.38e10T + 7.15e21T^{2} \) |
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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
−9.875219481800305076401652593780, −9.524537667122289270813623398167, −8.105738556299083999460822267977, −7.29394201018651160108354860232, −6.41621605500750235254369526197, −5.00788506799220357320333916247, −3.54825084968552247214876009451, −2.30990786195103037940662455571, −0.53629755298077713854372451436, 0,
0.53629755298077713854372451436, 2.30990786195103037940662455571, 3.54825084968552247214876009451, 5.00788506799220357320333916247, 6.41621605500750235254369526197, 7.29394201018651160108354860232, 8.105738556299083999460822267977, 9.524537667122289270813623398167, 9.875219481800305076401652593780
