| L(s) = 1 | − 0.947·2-s + 680.·3-s − 2.04e3·4-s − 1.09e4·5-s − 644.·6-s − 1.85e4·7-s + 3.87e3·8-s + 2.85e5·9-s + 1.03e4·10-s − 9.54e5·11-s − 1.39e6·12-s + 3.71e5·13-s + 1.75e4·14-s − 7.46e6·15-s + 4.18e6·16-s + 2.38e6·17-s − 2.70e5·18-s − 1.06e7·19-s + 2.24e7·20-s − 1.26e7·21-s + 9.04e5·22-s − 3.97e7·23-s + 2.63e6·24-s + 7.14e7·25-s − 3.51e5·26-s + 7.38e7·27-s + 3.79e7·28-s + ⋯ |
| L(s) = 1 | − 0.0209·2-s + 1.61·3-s − 0.999·4-s − 1.56·5-s − 0.0338·6-s − 0.417·7-s + 0.0418·8-s + 1.61·9-s + 0.0328·10-s − 1.78·11-s − 1.61·12-s + 0.277·13-s + 0.00873·14-s − 2.53·15-s + 0.998·16-s + 0.407·17-s − 0.0337·18-s − 0.983·19-s + 1.56·20-s − 0.674·21-s + 0.0374·22-s − 1.28·23-s + 0.0676·24-s + 1.46·25-s − 0.00580·26-s + 0.990·27-s + 0.417·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 - 3.71e5T \) |
| good | 2 | \( 1 + 0.947T + 2.04e3T^{2} \) |
| 3 | \( 1 - 680.T + 1.77e5T^{2} \) |
| 5 | \( 1 + 1.09e4T + 4.88e7T^{2} \) |
| 7 | \( 1 + 1.85e4T + 1.97e9T^{2} \) |
| 11 | \( 1 + 9.54e5T + 2.85e11T^{2} \) |
| 17 | \( 1 - 2.38e6T + 3.42e13T^{2} \) |
| 19 | \( 1 + 1.06e7T + 1.16e14T^{2} \) |
| 23 | \( 1 + 3.97e7T + 9.52e14T^{2} \) |
| 29 | \( 1 - 1.38e8T + 1.22e16T^{2} \) |
| 31 | \( 1 - 7.82e7T + 2.54e16T^{2} \) |
| 37 | \( 1 - 3.46e8T + 1.77e17T^{2} \) |
| 41 | \( 1 + 3.74e8T + 5.50e17T^{2} \) |
| 43 | \( 1 + 1.06e9T + 9.29e17T^{2} \) |
| 47 | \( 1 - 1.33e9T + 2.47e18T^{2} \) |
| 53 | \( 1 + 8.25e7T + 9.26e18T^{2} \) |
| 59 | \( 1 + 7.74e9T + 3.01e19T^{2} \) |
| 61 | \( 1 - 2.69e9T + 4.35e19T^{2} \) |
| 67 | \( 1 + 2.17e9T + 1.22e20T^{2} \) |
| 71 | \( 1 + 1.35e10T + 2.31e20T^{2} \) |
| 73 | \( 1 - 1.63e10T + 3.13e20T^{2} \) |
| 79 | \( 1 + 3.09e10T + 7.47e20T^{2} \) |
| 83 | \( 1 + 6.74e9T + 1.28e21T^{2} \) |
| 89 | \( 1 + 4.83e10T + 2.77e21T^{2} \) |
| 97 | \( 1 - 4.50e10T + 7.15e21T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−15.95759809596800065482897196388, −15.05014764402591526755854366674, −13.68526337206881135713484435132, −12.58046804417682116550390924994, −10.14375296467372551233153938489, −8.401061093425608052802118051375, −7.894631235267778561900753268585, −4.31319827103929694767851086470, −3.05377319735529544017020258813, 0,
3.05377319735529544017020258813, 4.31319827103929694767851086470, 7.894631235267778561900753268585, 8.401061093425608052802118051375, 10.14375296467372551233153938489, 12.58046804417682116550390924994, 13.68526337206881135713484435132, 15.05014764402591526755854366674, 15.95759809596800065482897196388
