| L(s) = 1 | + 32·2-s + 1.02e3·4-s + 8.08e3·5-s − 1.68e4·7-s + 3.27e4·8-s + 2.58e5·10-s − 9.29e5·11-s + 7.77e5·13-s − 5.37e5·14-s + 1.04e6·16-s − 3.10e6·17-s + 1.72e7·19-s + 8.27e6·20-s − 2.97e7·22-s + 4.28e7·23-s + 1.65e7·25-s + 2.48e7·26-s − 1.72e7·28-s + 8.18e7·29-s + 6.67e7·31-s + 3.35e7·32-s − 9.93e7·34-s − 1.35e8·35-s + 7.15e8·37-s + 5.50e8·38-s + 2.64e8·40-s − 7.58e8·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s + 1.15·5-s − 0.377·7-s + 0.353·8-s + 0.818·10-s − 1.73·11-s + 0.580·13-s − 0.267·14-s + 0.250·16-s − 0.530·17-s + 1.59·19-s + 0.578·20-s − 1.23·22-s + 1.38·23-s + 0.338·25-s + 0.410·26-s − 0.188·28-s + 0.740·29-s + 0.418·31-s + 0.176·32-s − 0.374·34-s − 0.437·35-s + 1.69·37-s + 1.12·38-s + 0.409·40-s − 1.02·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(4.316611525\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.316611525\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 32T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + 1.68e4T \) |
| good | 5 | \( 1 - 8.08e3T + 4.88e7T^{2} \) |
| 11 | \( 1 + 9.29e5T + 2.85e11T^{2} \) |
| 13 | \( 1 - 7.77e5T + 1.79e12T^{2} \) |
| 17 | \( 1 + 3.10e6T + 3.42e13T^{2} \) |
| 19 | \( 1 - 1.72e7T + 1.16e14T^{2} \) |
| 23 | \( 1 - 4.28e7T + 9.52e14T^{2} \) |
| 29 | \( 1 - 8.18e7T + 1.22e16T^{2} \) |
| 31 | \( 1 - 6.67e7T + 2.54e16T^{2} \) |
| 37 | \( 1 - 7.15e8T + 1.77e17T^{2} \) |
| 41 | \( 1 + 7.58e8T + 5.50e17T^{2} \) |
| 43 | \( 1 + 2.01e8T + 9.29e17T^{2} \) |
| 47 | \( 1 - 7.54e8T + 2.47e18T^{2} \) |
| 53 | \( 1 - 1.49e9T + 9.26e18T^{2} \) |
| 59 | \( 1 + 3.20e9T + 3.01e19T^{2} \) |
| 61 | \( 1 + 4.59e9T + 4.35e19T^{2} \) |
| 67 | \( 1 - 8.85e9T + 1.22e20T^{2} \) |
| 71 | \( 1 - 4.38e9T + 2.31e20T^{2} \) |
| 73 | \( 1 - 2.72e10T + 3.13e20T^{2} \) |
| 79 | \( 1 - 4.19e10T + 7.47e20T^{2} \) |
| 83 | \( 1 - 4.75e10T + 1.28e21T^{2} \) |
| 89 | \( 1 + 1.00e11T + 2.77e21T^{2} \) |
| 97 | \( 1 - 5.11e10T + 7.15e21T^{2} \) |
| 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
−11.20614510754262656306930154108, −10.29513182665390987938291921030, −9.373740516865019586543762032563, −7.930653994718993181905976907374, −6.68630713299453710849204241090, −5.63620547644849217873390713123, −4.90364582839867858514842731866, −3.16073704978073539658760730617, −2.35800384485245746749038770275, −0.925983219460122306855134422004,
0.925983219460122306855134422004, 2.35800384485245746749038770275, 3.16073704978073539658760730617, 4.90364582839867858514842731866, 5.63620547644849217873390713123, 6.68630713299453710849204241090, 7.930653994718993181905976907374, 9.373740516865019586543762032563, 10.29513182665390987938291921030, 11.20614510754262656306930154108
