| L(s) = 1 | + 0.262·2-s − 0.800·3-s − 1.93·4-s − 2.40·5-s − 0.209·6-s + 1.00·7-s − 1.03·8-s − 2.35·9-s − 0.631·10-s + 2.12·11-s + 1.54·12-s − 0.383·13-s + 0.264·14-s + 1.92·15-s + 3.59·16-s + 3.95·17-s − 0.618·18-s + 3.69·19-s + 4.65·20-s − 0.807·21-s + 0.555·22-s + 4.85·23-s + 0.824·24-s + 0.802·25-s − 0.100·26-s + 4.28·27-s − 1.94·28-s + ⋯ |
| L(s) = 1 | + 0.185·2-s − 0.461·3-s − 0.965·4-s − 1.07·5-s − 0.0856·6-s + 0.381·7-s − 0.364·8-s − 0.786·9-s − 0.199·10-s + 0.639·11-s + 0.446·12-s − 0.106·13-s + 0.0706·14-s + 0.497·15-s + 0.898·16-s + 0.959·17-s − 0.145·18-s + 0.848·19-s + 1.04·20-s − 0.176·21-s + 0.118·22-s + 1.01·23-s + 0.168·24-s + 0.160·25-s − 0.0197·26-s + 0.825·27-s − 0.368·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 619 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 619 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8354649248\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8354649248\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 619 | \( 1 - T \) |
| good | 2 | \( 1 - 0.262T + 2T^{2} \) |
| 3 | \( 1 + 0.800T + 3T^{2} \) |
| 5 | \( 1 + 2.40T + 5T^{2} \) |
| 7 | \( 1 - 1.00T + 7T^{2} \) |
| 11 | \( 1 - 2.12T + 11T^{2} \) |
| 13 | \( 1 + 0.383T + 13T^{2} \) |
| 17 | \( 1 - 3.95T + 17T^{2} \) |
| 19 | \( 1 - 3.69T + 19T^{2} \) |
| 23 | \( 1 - 4.85T + 23T^{2} \) |
| 29 | \( 1 + 3.08T + 29T^{2} \) |
| 31 | \( 1 - 0.275T + 31T^{2} \) |
| 37 | \( 1 + 3.06T + 37T^{2} \) |
| 41 | \( 1 - 2.32T + 41T^{2} \) |
| 43 | \( 1 - 8.80T + 43T^{2} \) |
| 47 | \( 1 - 1.90T + 47T^{2} \) |
| 53 | \( 1 - 8.51T + 53T^{2} \) |
| 59 | \( 1 - 0.748T + 59T^{2} \) |
| 61 | \( 1 - 8.47T + 61T^{2} \) |
| 67 | \( 1 + 7.23T + 67T^{2} \) |
| 71 | \( 1 - 13.5T + 71T^{2} \) |
| 73 | \( 1 + 3.23T + 73T^{2} \) |
| 79 | \( 1 + 5.96T + 79T^{2} \) |
| 83 | \( 1 - 2.05T + 83T^{2} \) |
| 89 | \( 1 + 6.75T + 89T^{2} \) |
| 97 | \( 1 - 1.61T + 97T^{2} \) |
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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
−10.81614740186200507545928119388, −9.645727291457475230866271763874, −8.828130690471232143400906658336, −8.039686255328189643070798256213, −7.20049225542117878473079438031, −5.79436110283242235410854804714, −5.09529225722850486678025381287, −4.05709891338098635631056534721, −3.18905624376828693971482150638, −0.806724464850011905546358673447,
0.806724464850011905546358673447, 3.18905624376828693971482150638, 4.05709891338098635631056534721, 5.09529225722850486678025381287, 5.79436110283242235410854804714, 7.20049225542117878473079438031, 8.039686255328189643070798256213, 8.828130690471232143400906658336, 9.645727291457475230866271763874, 10.81614740186200507545928119388
