| L(s) = 1 | + 8.51·2-s + 11.2·3-s + 40.5·4-s − 25·5-s + 95.6·6-s + 229.·7-s + 72.5·8-s − 116.·9-s − 212.·10-s + 502.·11-s + 455.·12-s + 169·13-s + 1.95e3·14-s − 280.·15-s − 678.·16-s − 453.·17-s − 995.·18-s − 1.89e3·19-s − 1.01e3·20-s + 2.57e3·21-s + 4.28e3·22-s − 1.31e3·23-s + 814.·24-s + 625·25-s + 1.43e3·26-s − 4.04e3·27-s + 9.30e3·28-s + ⋯ |
| L(s) = 1 | + 1.50·2-s + 0.720·3-s + 1.26·4-s − 0.447·5-s + 1.08·6-s + 1.77·7-s + 0.400·8-s − 0.481·9-s − 0.673·10-s + 1.25·11-s + 0.912·12-s + 0.277·13-s + 2.66·14-s − 0.322·15-s − 0.662·16-s − 0.380·17-s − 0.724·18-s − 1.20·19-s − 0.566·20-s + 1.27·21-s + 1.88·22-s − 0.517·23-s + 0.288·24-s + 0.200·25-s + 0.417·26-s − 1.06·27-s + 2.24·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(4.759297806\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.759297806\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + 25T \) |
| 13 | \( 1 - 169T \) |
| good | 2 | \( 1 - 8.51T + 32T^{2} \) |
| 3 | \( 1 - 11.2T + 243T^{2} \) |
| 7 | \( 1 - 229.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 502.T + 1.61e5T^{2} \) |
| 17 | \( 1 + 453.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 1.89e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 1.31e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 4.57e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 3.20e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 7.68e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.62e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.56e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 2.47e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.90e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 2.41e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 2.27e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 6.56e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 2.44e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 1.75e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 5.36e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 2.77e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 3.54e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 5.84e4T + 8.58e9T^{2} \) |
| 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
−14.27109288531381704751757820394, −13.03035839962444804336159727547, −11.63881436215678257225890576256, −11.25294440910647467145800069206, −8.922387800430152063532830502682, −7.941878476588159831681208925002, −6.25316346749667806010363266921, −4.71899630250145764116388999436, −3.77361088440411179797917789331, −2.04451279006606430019326417420,
2.04451279006606430019326417420, 3.77361088440411179797917789331, 4.71899630250145764116388999436, 6.25316346749667806010363266921, 7.941878476588159831681208925002, 8.922387800430152063532830502682, 11.25294440910647467145800069206, 11.63881436215678257225890576256, 13.03035839962444804336159727547, 14.27109288531381704751757820394
