| L(s) = 1 | + 38.1·2-s + 47.8·3-s + 944.·4-s − 109.·5-s + 1.82e3·6-s + 5.94e3·7-s + 1.65e4·8-s − 1.73e4·9-s − 4.18e3·10-s − 2.52e4·11-s + 4.52e4·12-s + 2.85e4·13-s + 2.27e5·14-s − 5.25e3·15-s + 1.46e5·16-s + 1.09e5·17-s − 6.63e5·18-s − 9.04e5·19-s − 1.03e5·20-s + 2.84e5·21-s − 9.62e5·22-s − 4.35e5·23-s + 7.90e5·24-s − 1.94e6·25-s + 1.09e6·26-s − 1.77e6·27-s + 5.61e6·28-s + ⋯ |
| L(s) = 1 | + 1.68·2-s + 0.341·3-s + 1.84·4-s − 0.0785·5-s + 0.575·6-s + 0.936·7-s + 1.42·8-s − 0.883·9-s − 0.132·10-s − 0.519·11-s + 0.629·12-s + 0.277·13-s + 1.57·14-s − 0.0268·15-s + 0.560·16-s + 0.317·17-s − 1.49·18-s − 1.59·19-s − 0.144·20-s + 0.319·21-s − 0.875·22-s − 0.324·23-s + 0.486·24-s − 0.993·25-s + 0.467·26-s − 0.642·27-s + 1.72·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(4.166673593\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.166673593\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 - 2.85e4T \) |
| good | 2 | \( 1 - 38.1T + 512T^{2} \) |
| 3 | \( 1 - 47.8T + 1.96e4T^{2} \) |
| 5 | \( 1 + 109.T + 1.95e6T^{2} \) |
| 7 | \( 1 - 5.94e3T + 4.03e7T^{2} \) |
| 11 | \( 1 + 2.52e4T + 2.35e9T^{2} \) |
| 17 | \( 1 - 1.09e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 9.04e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 4.35e5T + 1.80e12T^{2} \) |
| 29 | \( 1 - 6.44e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 6.62e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 4.14e6T + 1.29e14T^{2} \) |
| 41 | \( 1 - 1.49e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 4.01e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 6.30e6T + 1.11e15T^{2} \) |
| 53 | \( 1 - 1.53e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 1.52e8T + 8.66e15T^{2} \) |
| 61 | \( 1 - 8.66e7T + 1.16e16T^{2} \) |
| 67 | \( 1 + 1.01e8T + 2.72e16T^{2} \) |
| 71 | \( 1 - 4.13e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 3.14e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 2.00e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 6.34e7T + 1.86e17T^{2} \) |
| 89 | \( 1 - 3.47e7T + 3.50e17T^{2} \) |
| 97 | \( 1 + 1.25e9T + 7.60e17T^{2} \) |
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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
−17.50090066173110072962276206492, −15.69377707630380753340650373156, −14.56582926509755645164753612929, −13.72040950727549817230085663465, −12.22243839575002202312497247507, −10.96755344068652021440322682103, −8.172037666415081195687127974513, −5.99103095338263178456271172566, −4.40347901934242206136518659056, −2.53158953959929218221181292797,
2.53158953959929218221181292797, 4.40347901934242206136518659056, 5.99103095338263178456271172566, 8.172037666415081195687127974513, 10.96755344068652021440322682103, 12.22243839575002202312497247507, 13.72040950727549817230085663465, 14.56582926509755645164753612929, 15.69377707630380753340650373156, 17.50090066173110072962276206492
