| L(s) = 1 | − 70.8·2-s − 19.0·3-s + 2.96e3·4-s − 1.18e4·5-s + 1.35e3·6-s − 6.22e4·7-s − 6.49e4·8-s − 1.76e5·9-s + 8.41e5·10-s + 5.97e5·11-s − 5.65e4·12-s − 3.71e5·13-s + 4.40e6·14-s + 2.26e5·15-s − 1.47e6·16-s + 8.43e6·17-s + 1.25e7·18-s − 1.00e7·19-s − 3.52e7·20-s + 1.18e6·21-s − 4.23e7·22-s + 2.22e7·23-s + 1.23e6·24-s + 9.24e7·25-s + 2.62e7·26-s + 6.75e6·27-s − 1.84e8·28-s + ⋯ |
| L(s) = 1 | − 1.56·2-s − 0.0453·3-s + 1.44·4-s − 1.70·5-s + 0.0709·6-s − 1.39·7-s − 0.700·8-s − 0.997·9-s + 2.66·10-s + 1.11·11-s − 0.0656·12-s − 0.277·13-s + 2.18·14-s + 0.0771·15-s − 0.351·16-s + 1.44·17-s + 1.56·18-s − 0.935·19-s − 2.46·20-s + 0.0634·21-s − 1.75·22-s + 0.720·23-s + 0.0317·24-s + 1.89·25-s + 0.433·26-s + 0.0905·27-s − 2.02·28-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
| \(L(6)\) |
\(\approx\) |
\(0.2675874129\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2675874129\) |
| \(L(\frac{13}{2})\) |
|
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 | \( 1 + 70.8T + 2.04e3T^{2} \) |
| 3 | \( 1 + 19.0T + 1.77e5T^{2} \) |
| 5 | \( 1 + 1.18e4T + 4.88e7T^{2} \) |
| 7 | \( 1 + 6.22e4T + 1.97e9T^{2} \) |
| 11 | \( 1 - 5.97e5T + 2.85e11T^{2} \) |
| 17 | \( 1 - 8.43e6T + 3.42e13T^{2} \) |
| 19 | \( 1 + 1.00e7T + 1.16e14T^{2} \) |
| 23 | \( 1 - 2.22e7T + 9.52e14T^{2} \) |
| 29 | \( 1 + 2.34e7T + 1.22e16T^{2} \) |
| 31 | \( 1 + 1.58e8T + 2.54e16T^{2} \) |
| 37 | \( 1 - 2.36e8T + 1.77e17T^{2} \) |
| 41 | \( 1 + 7.38e8T + 5.50e17T^{2} \) |
| 43 | \( 1 - 1.53e8T + 9.29e17T^{2} \) |
| 47 | \( 1 + 1.40e9T + 2.47e18T^{2} \) |
| 53 | \( 1 + 1.36e9T + 9.26e18T^{2} \) |
| 59 | \( 1 + 3.06e9T + 3.01e19T^{2} \) |
| 61 | \( 1 - 7.81e9T + 4.35e19T^{2} \) |
| 67 | \( 1 + 2.17e9T + 1.22e20T^{2} \) |
| 71 | \( 1 - 2.03e10T + 2.31e20T^{2} \) |
| 73 | \( 1 - 2.68e10T + 3.13e20T^{2} \) |
| 79 | \( 1 + 1.22e10T + 7.47e20T^{2} \) |
| 83 | \( 1 - 6.41e10T + 1.28e21T^{2} \) |
| 89 | \( 1 + 1.49e10T + 2.77e21T^{2} \) |
| 97 | \( 1 + 1.11e11T + 7.15e21T^{2} \) |
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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
−16.90508903422816934111202247522, −16.37338610686192155077793029883, −14.90436685332168044499224481545, −12.27761701415296209437028099233, −11.10759857793172238020589644325, −9.434810230882670977245120340156, −8.192530546688265120188789904710, −6.80519869419596659311116999305, −3.42584485121743564094227780078, −0.50603081578648012996086655723,
0.50603081578648012996086655723, 3.42584485121743564094227780078, 6.80519869419596659311116999305, 8.192530546688265120188789904710, 9.434810230882670977245120340156, 11.10759857793172238020589644325, 12.27761701415296209437028099233, 14.90436685332168044499224481545, 16.37338610686192155077793029883, 16.90508903422816934111202247522
