| L(s) = 1 | + 21.9·2-s − 1.56e3·4-s + 886.·5-s + 4.42e4·7-s − 7.94e4·8-s + 1.94e4·10-s + 4.80e5·11-s + 7.29e5·13-s + 9.73e5·14-s + 1.45e6·16-s − 6.80e6·17-s − 1.56e7·19-s − 1.38e6·20-s + 1.05e7·22-s + 3.18e7·23-s − 4.80e7·25-s + 1.60e7·26-s − 6.92e7·28-s − 2.05e7·29-s − 3.08e8·31-s + 1.94e8·32-s − 1.49e8·34-s + 3.92e7·35-s + 4.28e8·37-s − 3.43e8·38-s − 7.04e7·40-s + 1.00e9·41-s + ⋯ |
| L(s) = 1 | + 0.486·2-s − 0.763·4-s + 0.126·5-s + 0.995·7-s − 0.857·8-s + 0.0616·10-s + 0.899·11-s + 0.544·13-s + 0.483·14-s + 0.346·16-s − 1.16·17-s − 1.44·19-s − 0.0968·20-s + 0.437·22-s + 1.03·23-s − 0.983·25-s + 0.264·26-s − 0.760·28-s − 0.185·29-s − 1.93·31-s + 1.02·32-s − 0.565·34-s + 0.126·35-s + 1.01·37-s − 0.703·38-s − 0.108·40-s + 1.36·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 261 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 261 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 29 | \( 1 + 2.05e7T \) |
| good | 2 | \( 1 - 21.9T + 2.04e3T^{2} \) |
| 5 | \( 1 - 886.T + 4.88e7T^{2} \) |
| 7 | \( 1 - 4.42e4T + 1.97e9T^{2} \) |
| 11 | \( 1 - 4.80e5T + 2.85e11T^{2} \) |
| 13 | \( 1 - 7.29e5T + 1.79e12T^{2} \) |
| 17 | \( 1 + 6.80e6T + 3.42e13T^{2} \) |
| 19 | \( 1 + 1.56e7T + 1.16e14T^{2} \) |
| 23 | \( 1 - 3.18e7T + 9.52e14T^{2} \) |
| 31 | \( 1 + 3.08e8T + 2.54e16T^{2} \) |
| 37 | \( 1 - 4.28e8T + 1.77e17T^{2} \) |
| 41 | \( 1 - 1.00e9T + 5.50e17T^{2} \) |
| 43 | \( 1 - 3.53e8T + 9.29e17T^{2} \) |
| 47 | \( 1 - 2.87e9T + 2.47e18T^{2} \) |
| 53 | \( 1 - 1.08e9T + 9.26e18T^{2} \) |
| 59 | \( 1 + 9.69e9T + 3.01e19T^{2} \) |
| 61 | \( 1 - 2.73e9T + 4.35e19T^{2} \) |
| 67 | \( 1 - 5.53e9T + 1.22e20T^{2} \) |
| 71 | \( 1 + 1.51e10T + 2.31e20T^{2} \) |
| 73 | \( 1 - 2.05e10T + 3.13e20T^{2} \) |
| 79 | \( 1 + 2.27e10T + 7.47e20T^{2} \) |
| 83 | \( 1 - 6.37e9T + 1.28e21T^{2} \) |
| 89 | \( 1 - 2.92e10T + 2.77e21T^{2} \) |
| 97 | \( 1 + 1.24e11T + 7.15e21T^{2} \) |
| 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
−9.193380070153092567506553894220, −8.891086121975628428061438225021, −7.73306671845372209079004208810, −6.41908165025524001560606537711, −5.52559679770951511043775829856, −4.36998122810509496299144902663, −3.92639801153879941463072422267, −2.34046001603312245481203303799, −1.23433456638183917531514212723, 0,
1.23433456638183917531514212723, 2.34046001603312245481203303799, 3.92639801153879941463072422267, 4.36998122810509496299144902663, 5.52559679770951511043775829856, 6.41908165025524001560606537711, 7.73306671845372209079004208810, 8.891086121975628428061438225021, 9.193380070153092567506553894220
