| L(s) = 1 | − 6·2-s + 4·3-s + 4·4-s + 54·5-s − 24·6-s + 248·7-s + 168·8-s − 227·9-s − 324·10-s + 204·11-s + 16·12-s − 370·13-s − 1.48e3·14-s + 216·15-s − 1.13e3·16-s + 1.55e3·17-s + 1.36e3·18-s + 361·19-s + 216·20-s + 992·21-s − 1.22e3·22-s − 408·23-s + 672·24-s − 209·25-s + 2.22e3·26-s − 1.88e3·27-s + 992·28-s + ⋯ |
| L(s) = 1 | − 1.06·2-s + 0.256·3-s + 1/8·4-s + 0.965·5-s − 0.272·6-s + 1.91·7-s + 0.928·8-s − 0.934·9-s − 1.02·10-s + 0.508·11-s + 0.0320·12-s − 0.607·13-s − 2.02·14-s + 0.247·15-s − 1.10·16-s + 1.30·17-s + 0.990·18-s + 0.229·19-s + 0.120·20-s + 0.490·21-s − 0.539·22-s − 0.160·23-s + 0.238·24-s − 0.0668·25-s + 0.644·26-s − 0.496·27-s + 0.239·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.062731615\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.062731615\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 19 | \( 1 - p^{2} T \) |
| good | 2 | \( 1 + 3 p T + p^{5} T^{2} \) |
| 3 | \( 1 - 4 T + p^{5} T^{2} \) |
| 5 | \( 1 - 54 T + p^{5} T^{2} \) |
| 7 | \( 1 - 248 T + p^{5} T^{2} \) |
| 11 | \( 1 - 204 T + p^{5} T^{2} \) |
| 13 | \( 1 + 370 T + p^{5} T^{2} \) |
| 17 | \( 1 - 1554 T + p^{5} T^{2} \) |
| 23 | \( 1 + 408 T + p^{5} T^{2} \) |
| 29 | \( 1 - 6174 T + p^{5} T^{2} \) |
| 31 | \( 1 + 7840 T + p^{5} T^{2} \) |
| 37 | \( 1 + 5146 T + p^{5} T^{2} \) |
| 41 | \( 1 + 7830 T + p^{5} T^{2} \) |
| 43 | \( 1 + 12532 T + p^{5} T^{2} \) |
| 47 | \( 1 - 2592 T + p^{5} T^{2} \) |
| 53 | \( 1 + 20778 T + p^{5} T^{2} \) |
| 59 | \( 1 - 18972 T + p^{5} T^{2} \) |
| 61 | \( 1 + 18418 T + p^{5} T^{2} \) |
| 67 | \( 1 + 11548 T + p^{5} T^{2} \) |
| 71 | \( 1 + 72984 T + p^{5} T^{2} \) |
| 73 | \( 1 - 59114 T + p^{5} T^{2} \) |
| 79 | \( 1 + 44752 T + p^{5} T^{2} \) |
| 83 | \( 1 + 27660 T + p^{5} T^{2} \) |
| 89 | \( 1 - 20730 T + p^{5} T^{2} \) |
| 97 | \( 1 - 14018 T + p^{5} T^{2} \) |
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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
−17.49414630087554256041751403239, −16.91100644697927216903062101811, −14.51563716179166835545513646687, −13.97402738671033088547095336469, −11.67921123654098140212504354651, −10.24396183724084409067397000643, −8.878126809240744831028403552950, −7.79537978775415403275626405600, −5.20617286449192399233955357641, −1.64374547080511847774863094259,
1.64374547080511847774863094259, 5.20617286449192399233955357641, 7.79537978775415403275626405600, 8.878126809240744831028403552950, 10.24396183724084409067397000643, 11.67921123654098140212504354651, 13.97402738671033088547095336469, 14.51563716179166835545513646687, 16.91100644697927216903062101811, 17.49414630087554256041751403239
