| L(s) = 1 | + (−0.618 − 1.90i)2-s + (−6.71 + 4.88i)3-s + (−3.23 + 2.35i)4-s + (13.4 + 9.76i)6-s + 11.9·7-s + (6.47 + 4.70i)8-s + (12.9 − 39.9i)9-s + (16.7 + 51.6i)11-s + (10.2 − 31.5i)12-s + (20.9 − 64.4i)13-s + (−7.39 − 22.7i)14-s + (4.94 − 15.2i)16-s + (−8.59 − 6.24i)17-s − 83.9·18-s + (−18.5 − 13.5i)19-s + ⋯ |
| L(s) = 1 | + (−0.218 − 0.672i)2-s + (−1.29 + 0.939i)3-s + (−0.404 + 0.293i)4-s + (0.914 + 0.664i)6-s + 0.645·7-s + (0.286 + 0.207i)8-s + (0.480 − 1.47i)9-s + (0.460 + 1.41i)11-s + (0.246 − 0.760i)12-s + (0.447 − 1.37i)13-s + (−0.141 − 0.434i)14-s + (0.0772 − 0.237i)16-s + (−0.122 − 0.0891i)17-s − 1.09·18-s + (−0.224 − 0.163i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 250 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.455 - 0.890i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 250 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.455 - 0.890i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.316107 + 0.517109i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.316107 + 0.517109i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.618 + 1.90i)T \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + (6.71 - 4.88i)T + (8.34 - 25.6i)T^{2} \) |
| 7 | \( 1 - 11.9T + 343T^{2} \) |
| 11 | \( 1 + (-16.7 - 51.6i)T + (-1.07e3 + 782. i)T^{2} \) |
| 13 | \( 1 + (-20.9 + 64.4i)T + (-1.77e3 - 1.29e3i)T^{2} \) |
| 17 | \( 1 + (8.59 + 6.24i)T + (1.51e3 + 4.67e3i)T^{2} \) |
| 19 | \( 1 + (18.5 + 13.5i)T + (2.11e3 + 6.52e3i)T^{2} \) |
| 23 | \( 1 + (-45.4 - 140. i)T + (-9.84e3 + 7.15e3i)T^{2} \) |
| 29 | \( 1 + (166. - 121. i)T + (7.53e3 - 2.31e4i)T^{2} \) |
| 31 | \( 1 + (-70.9 - 51.5i)T + (9.20e3 + 2.83e4i)T^{2} \) |
| 37 | \( 1 + (127. - 393. i)T + (-4.09e4 - 2.97e4i)T^{2} \) |
| 41 | \( 1 + (20.9 - 64.3i)T + (-5.57e4 - 4.05e4i)T^{2} \) |
| 43 | \( 1 + 471.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (-53.4 + 38.8i)T + (3.20e4 - 9.87e4i)T^{2} \) |
| 53 | \( 1 + (-255. + 185. i)T + (4.60e4 - 1.41e5i)T^{2} \) |
| 59 | \( 1 + (31.3 - 96.3i)T + (-1.66e5 - 1.20e5i)T^{2} \) |
| 61 | \( 1 + (125. + 385. i)T + (-1.83e5 + 1.33e5i)T^{2} \) |
| 67 | \( 1 + (425. + 308. i)T + (9.29e4 + 2.86e5i)T^{2} \) |
| 71 | \( 1 + (-386. + 280. i)T + (1.10e5 - 3.40e5i)T^{2} \) |
| 73 | \( 1 + (-264. - 815. i)T + (-3.14e5 + 2.28e5i)T^{2} \) |
| 79 | \( 1 + (444. - 322. i)T + (1.52e5 - 4.68e5i)T^{2} \) |
| 83 | \( 1 + (-232. - 169. i)T + (1.76e5 + 5.43e5i)T^{2} \) |
| 89 | \( 1 + (12.3 + 37.9i)T + (-5.70e5 + 4.14e5i)T^{2} \) |
| 97 | \( 1 + (833. - 605. i)T + (2.82e5 - 8.68e5i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.65881636332235965650743175777, −10.99260727311940229972259452559, −10.17077617291388288120641946119, −9.520478843970382410709517155349, −8.182649435778756152928366386462, −6.84099289432263308206392499347, −5.35727673889344020993095979882, −4.75904459517460167934848188134, −3.50474614082634548632521361415, −1.42015569879401814130074604148,
0.33819275537822039371571172382, 1.65118809403838743887997952946, 4.22043468190699007724740408254, 5.53480154067957260945344033925, 6.29917445547807911518291997972, 7.02392613189338579829953290987, 8.216812338144935410800777692422, 9.074172134806216855438541010358, 10.72216828499353197186171295261, 11.33176309870465686291007345521
