| L(s) = 1 | − 3.63·2-s + 5.33·3-s + 5.21·4-s − 11.0·5-s − 19.3·6-s + 10.1·8-s + 1.48·9-s + 39.9·10-s − 11.7·11-s + 27.8·12-s + 13·13-s − 58.7·15-s − 78.5·16-s + 102.·17-s − 5.41·18-s − 43.2·19-s − 57.3·20-s + 42.5·22-s − 25.9·23-s + 54.1·24-s − 3.93·25-s − 47.2·26-s − 136.·27-s − 272.·29-s + 213.·30-s + 121.·31-s + 204.·32-s + ⋯ |
| L(s) = 1 | − 1.28·2-s + 1.02·3-s + 0.651·4-s − 0.984·5-s − 1.31·6-s + 0.448·8-s + 0.0551·9-s + 1.26·10-s − 0.320·11-s + 0.668·12-s + 0.277·13-s − 1.01·15-s − 1.22·16-s + 1.46·17-s − 0.0708·18-s − 0.522·19-s − 0.640·20-s + 0.412·22-s − 0.235·23-s + 0.460·24-s − 0.0314·25-s − 0.356·26-s − 0.970·27-s − 1.74·29-s + 1.29·30-s + 0.702·31-s + 1.12·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.9247416843\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9247416843\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 13 | \( 1 - 13T \) |
| good | 2 | \( 1 + 3.63T + 8T^{2} \) |
| 3 | \( 1 - 5.33T + 27T^{2} \) |
| 5 | \( 1 + 11.0T + 125T^{2} \) |
| 11 | \( 1 + 11.7T + 1.33e3T^{2} \) |
| 17 | \( 1 - 102.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 43.2T + 6.85e3T^{2} \) |
| 23 | \( 1 + 25.9T + 1.21e4T^{2} \) |
| 29 | \( 1 + 272.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 121.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 168.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 451.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 94.6T + 7.95e4T^{2} \) |
| 47 | \( 1 + 50.6T + 1.03e5T^{2} \) |
| 53 | \( 1 + 398.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 686.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 75.3T + 2.26e5T^{2} \) |
| 67 | \( 1 + 336.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 427.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 134.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 253.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 193.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 995.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 761.T + 9.12e5T^{2} \) |
| 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
−9.823751801872555041714289848094, −9.258326842581366197674118979997, −8.280232928973618892998601765609, −7.908164301493247928072715170266, −7.31663139948896536721116654219, −5.79739860777240498851366873930, −4.27571261878339588144802216775, −3.35637337814673355949900831068, −2.09615915652645558356068895376, −0.64578977229240372491271704913,
0.64578977229240372491271704913, 2.09615915652645558356068895376, 3.35637337814673355949900831068, 4.27571261878339588144802216775, 5.79739860777240498851366873930, 7.31663139948896536721116654219, 7.908164301493247928072715170266, 8.280232928973618892998601765609, 9.258326842581366197674118979997, 9.823751801872555041714289848094
