| L(s) = 1 | − 0.396·2-s + 11.4·3-s − 31.8·4-s − 4.55·6-s − 8.04·7-s + 25.3·8-s − 110.·9-s + 335.·11-s − 366.·12-s + 169·13-s + 3.19·14-s + 1.00e3·16-s − 38.1·17-s + 43.9·18-s + 1.17e3·19-s − 92.5·21-s − 133.·22-s − 1.67e3·23-s + 291.·24-s − 67.0·26-s − 4.06e3·27-s + 256.·28-s − 6.70e3·29-s + 7.85e3·31-s − 1.21e3·32-s + 3.86e3·33-s + 15.1·34-s + ⋯ |
| L(s) = 1 | − 0.0701·2-s + 0.737·3-s − 0.995·4-s − 0.0517·6-s − 0.0620·7-s + 0.139·8-s − 0.456·9-s + 0.837·11-s − 0.733·12-s + 0.277·13-s + 0.00435·14-s + 0.985·16-s − 0.0320·17-s + 0.0319·18-s + 0.749·19-s − 0.0457·21-s − 0.0586·22-s − 0.661·23-s + 0.103·24-s − 0.0194·26-s − 1.07·27-s + 0.0617·28-s − 1.48·29-s + 1.46·31-s − 0.208·32-s + 0.617·33-s + 0.00224·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 13 | \( 1 - 169T \) |
| good | 2 | \( 1 + 0.396T + 32T^{2} \) |
| 3 | \( 1 - 11.4T + 243T^{2} \) |
| 7 | \( 1 + 8.04T + 1.68e4T^{2} \) |
| 11 | \( 1 - 335.T + 1.61e5T^{2} \) |
| 17 | \( 1 + 38.1T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.17e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 1.67e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 6.70e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 7.85e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.71e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.00e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.08e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.29e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 2.63e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 4.95e3T + 7.14e8T^{2} \) |
| 61 | \( 1 + 4.37e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 9.17e3T + 1.35e9T^{2} \) |
| 71 | \( 1 + 1.30e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 4.17e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 4.98e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 8.61e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 6.72e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.75e5T + 8.58e9T^{2} \) |
| 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
−9.919178766219679368280239677961, −9.336014423258543631683780896661, −8.481168215038057377185864285940, −7.79893335464543827104526362177, −6.36086181805407351487726758779, −5.22063237338730465216192313910, −3.99399976567016537951048892975, −3.13999922285196104057989101240, −1.50641492870067503442889471003, 0,
1.50641492870067503442889471003, 3.13999922285196104057989101240, 3.99399976567016537951048892975, 5.22063237338730465216192313910, 6.36086181805407351487726758779, 7.79893335464543827104526362177, 8.481168215038057377185864285940, 9.336014423258543631683780896661, 9.919178766219679368280239677961
