| L(s) = 1 | − 25·5-s − 53.1·7-s − 585.·11-s − 1.08e3·13-s − 1.59e3·17-s − 2.41e3·19-s − 2.67e3·23-s + 625·25-s + 8.43e3·29-s + 5.02e3·31-s + 1.32e3·35-s + 1.82e3·37-s + 7.97e3·41-s − 1.05e4·43-s − 2.93e4·47-s − 1.39e4·49-s + 2.30e4·53-s + 1.46e4·55-s + 2.21e3·59-s + 2.19e4·61-s + 2.71e4·65-s + 410.·67-s − 5.36e4·71-s − 1.75e4·73-s + 3.11e4·77-s − 5.95e4·79-s + 2.29e4·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.409·7-s − 1.45·11-s − 1.78·13-s − 1.33·17-s − 1.53·19-s − 1.05·23-s + 0.200·25-s + 1.86·29-s + 0.938·31-s + 0.183·35-s + 0.219·37-s + 0.740·41-s − 0.870·43-s − 1.94·47-s − 0.831·49-s + 1.12·53-s + 0.652·55-s + 0.0828·59-s + 0.754·61-s + 0.796·65-s + 0.0111·67-s − 1.26·71-s − 0.385·73-s + 0.598·77-s − 1.07·79-s + 0.365·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.2466327735\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2466327735\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + 25T \) |
| good | 7 | \( 1 + 53.1T + 1.68e4T^{2} \) |
| 11 | \( 1 + 585.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 1.08e3T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.59e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 2.41e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 2.67e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 8.43e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 5.02e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.82e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 7.97e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.05e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 2.93e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.30e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 2.21e3T + 7.14e8T^{2} \) |
| 61 | \( 1 - 2.19e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 410.T + 1.35e9T^{2} \) |
| 71 | \( 1 + 5.36e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 1.75e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 5.95e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 2.29e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 5.67e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.51e5T + 8.58e9T^{2} \) |
| 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
−9.960107110836252982392051314527, −8.575820325690600994297719390437, −8.032089029128746149208952407447, −7.01181684544866556646615656032, −6.25135251890943609837483653391, −4.89160465961261275792169436481, −4.39237762429918622218785000345, −2.83466990079058996364521403339, −2.21960959959677683058713999778, −0.21223090928362409526216257839,
0.21223090928362409526216257839, 2.21960959959677683058713999778, 2.83466990079058996364521403339, 4.39237762429918622218785000345, 4.89160465961261275792169436481, 6.25135251890943609837483653391, 7.01181684544866556646615656032, 8.032089029128746149208952407447, 8.575820325690600994297719390437, 9.960107110836252982392051314527
