| L(s) = 1 | − 13.3·2-s − 333.·4-s − 1.92e3·5-s + 2.40e3·7-s + 1.12e4·8-s + 2.56e4·10-s + 9.01e4·11-s − 3.19e3·13-s − 3.20e4·14-s + 1.98e4·16-s − 1.16e5·17-s − 1.42e5·19-s + 6.41e5·20-s − 1.20e6·22-s − 1.27e6·23-s + 1.74e6·25-s + 4.27e4·26-s − 8.00e5·28-s + 1.42e6·29-s + 9.67e6·31-s − 6.04e6·32-s + 1.55e6·34-s − 4.61e6·35-s − 8.67e6·37-s + 1.90e6·38-s − 2.17e7·40-s − 1.32e7·41-s + ⋯ |
| L(s) = 1 | − 0.590·2-s − 0.651·4-s − 1.37·5-s + 0.377·7-s + 0.975·8-s + 0.812·10-s + 1.85·11-s − 0.0310·13-s − 0.223·14-s + 0.0756·16-s − 0.338·17-s − 0.250·19-s + 0.895·20-s − 1.09·22-s − 0.949·23-s + 0.891·25-s + 0.0183·26-s − 0.246·28-s + 0.375·29-s + 1.88·31-s − 1.01·32-s + 0.199·34-s − 0.519·35-s − 0.761·37-s + 0.148·38-s − 1.34·40-s − 0.732·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 - 2.40e3T \) |
| good | 2 | \( 1 + 13.3T + 512T^{2} \) |
| 5 | \( 1 + 1.92e3T + 1.95e6T^{2} \) |
| 11 | \( 1 - 9.01e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 3.19e3T + 1.06e10T^{2} \) |
| 17 | \( 1 + 1.16e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 1.42e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 1.27e6T + 1.80e12T^{2} \) |
| 29 | \( 1 - 1.42e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 9.67e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 8.67e6T + 1.29e14T^{2} \) |
| 41 | \( 1 + 1.32e7T + 3.27e14T^{2} \) |
| 43 | \( 1 + 2.97e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 1.07e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 7.07e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 6.40e6T + 8.66e15T^{2} \) |
| 61 | \( 1 - 1.69e8T + 1.16e16T^{2} \) |
| 67 | \( 1 + 1.16e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + 1.44e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 1.60e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 4.89e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 8.31e7T + 1.86e17T^{2} \) |
| 89 | \( 1 + 2.08e6T + 3.50e17T^{2} \) |
| 97 | \( 1 + 3.15e8T + 7.60e17T^{2} \) |
| 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
−12.15734158020072350618092760625, −11.47651951040733311141169862289, −10.04894910797532452527094488332, −8.760110974183384334343837870780, −8.056491474823837420230041668973, −6.71042080146686113068152406349, −4.58259433621368708441529712567, −3.77246288940839410234400663429, −1.30422893311466732309819377029, 0,
1.30422893311466732309819377029, 3.77246288940839410234400663429, 4.58259433621368708441529712567, 6.71042080146686113068152406349, 8.056491474823837420230041668973, 8.760110974183384334343837870780, 10.04894910797532452527094488332, 11.47651951040733311141169862289, 12.15734158020072350618092760625
