| L(s) = 1 | + 4.68·2-s + 13.5·3-s − 10.0·4-s + 15.4·5-s + 63.2·6-s + 12.5·7-s − 197.·8-s − 60.4·9-s + 72.2·10-s − 271.·11-s − 136.·12-s + 169·13-s + 58.6·14-s + 208.·15-s − 599.·16-s + 1.71e3·17-s − 282.·18-s + 2.23e3·19-s − 155.·20-s + 169.·21-s − 1.27e3·22-s − 966.·23-s − 2.66e3·24-s − 2.88e3·25-s + 791.·26-s − 4.09e3·27-s − 126.·28-s + ⋯ |
| L(s) = 1 | + 0.827·2-s + 0.866·3-s − 0.315·4-s + 0.275·5-s + 0.717·6-s + 0.0966·7-s − 1.08·8-s − 0.248·9-s + 0.228·10-s − 0.676·11-s − 0.273·12-s + 0.277·13-s + 0.0800·14-s + 0.239·15-s − 0.585·16-s + 1.43·17-s − 0.205·18-s + 1.42·19-s − 0.0870·20-s + 0.0838·21-s − 0.559·22-s − 0.380·23-s − 0.943·24-s − 0.923·25-s + 0.229·26-s − 1.08·27-s − 0.0304·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.970766487\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.970766487\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 - 169T \) |
| good | 2 | \( 1 - 4.68T + 32T^{2} \) |
| 3 | \( 1 - 13.5T + 243T^{2} \) |
| 5 | \( 1 - 15.4T + 3.12e3T^{2} \) |
| 7 | \( 1 - 12.5T + 1.68e4T^{2} \) |
| 11 | \( 1 + 271.T + 1.61e5T^{2} \) |
| 17 | \( 1 - 1.71e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 2.23e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 966.T + 6.43e6T^{2} \) |
| 29 | \( 1 - 4.91e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 2.31e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.34e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.64e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 9.52e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.21e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 3.06e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 3.72e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 3.05e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 4.41e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 1.68e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 1.31e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 1.98e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 2.22e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 3.87e3T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.55e4T + 8.58e9T^{2} \) |
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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
−18.95090969706949807512924302484, −17.66913518648432377916031590885, −15.69482000589633183408237187765, −14.24759147369241459631069285707, −13.66472340954359445757121044428, −12.07527670944691740524879536936, −9.733414434890861993842738002440, −8.150633012489459740691216890730, −5.47954237252124404334651225374, −3.24544101928119443496424399361,
3.24544101928119443496424399361, 5.47954237252124404334651225374, 8.150633012489459740691216890730, 9.733414434890861993842738002440, 12.07527670944691740524879536936, 13.66472340954359445757121044428, 14.24759147369241459631069285707, 15.69482000589633183408237187765, 17.66913518648432377916031590885, 18.95090969706949807512924302484
