| L(s) = 1 | − 27.0·3-s − 61.5·5-s + 221.·7-s + 490.·9-s + 180.·11-s − 221.·13-s + 1.66e3·15-s + 856.·17-s − 1.36e3·19-s − 5.98e3·21-s − 2.39e3·23-s + 664.·25-s − 6.69e3·27-s − 7.75e3·29-s − 677.·31-s − 4.88e3·33-s − 1.36e4·35-s + 9.44e3·37-s + 6.00e3·39-s + 2.00e3·41-s + 2.29e3·43-s − 3.01e4·45-s + 1.45e4·47-s + 3.20e4·49-s − 2.31e4·51-s + 3.02e4·53-s − 1.11e4·55-s + ⋯ |
| L(s) = 1 | − 1.73·3-s − 1.10·5-s + 1.70·7-s + 2.01·9-s + 0.449·11-s − 0.364·13-s + 1.91·15-s + 0.718·17-s − 0.868·19-s − 2.96·21-s − 0.945·23-s + 0.212·25-s − 1.76·27-s − 1.71·29-s − 0.126·31-s − 0.781·33-s − 1.87·35-s + 1.13·37-s + 0.632·39-s + 0.186·41-s + 0.189·43-s − 2.22·45-s + 0.958·47-s + 1.90·49-s − 1.24·51-s + 1.47·53-s − 0.495·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1028 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1028 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 257 | \( 1 - 6.60e4T \) |
| good | 3 | \( 1 + 27.0T + 243T^{2} \) |
| 5 | \( 1 + 61.5T + 3.12e3T^{2} \) |
| 7 | \( 1 - 221.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 180.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 221.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 856.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 1.36e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 2.39e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 7.75e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 677.T + 2.86e7T^{2} \) |
| 37 | \( 1 - 9.44e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 2.00e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 2.29e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.45e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 3.02e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 2.66e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 3.26e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 2.19e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 1.06e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 6.94e3T + 2.07e9T^{2} \) |
| 79 | \( 1 + 6.76e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 4.68e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 6.55e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 5.11e4T + 8.58e9T^{2} \) |
| 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
−8.622435068972547170772810898410, −7.66234148854979759856679468804, −7.29120973678331222516266293411, −5.99984171517737430936181419357, −5.39282637510614930172965592875, −4.38799804958309552234657475700, −4.03048559880945648180364199715, −1.95604825675658568115066268355, −0.954907084429110741567593990787, 0,
0.954907084429110741567593990787, 1.95604825675658568115066268355, 4.03048559880945648180364199715, 4.38799804958309552234657475700, 5.39282637510614930172965592875, 5.99984171517737430936181419357, 7.29120973678331222516266293411, 7.66234148854979759856679468804, 8.622435068972547170772810898410
