| L(s) = 1 | + (−1.20 − 3.70i)2-s + (−5.80 + 4.22i)4-s + (3.34 + 10.6i)5-s + 6.27·7-s + (−2.58 − 1.87i)8-s + (35.5 − 25.2i)10-s + (−13.6 − 41.9i)11-s + (−13.2 + 40.7i)13-s + (−7.56 − 23.2i)14-s + (−21.5 + 66.4i)16-s + (−81.6 − 59.3i)17-s + (−65.1 − 47.3i)19-s + (−64.4 − 47.8i)20-s + (−139. + 101. i)22-s + (−48.5 − 149. i)23-s + ⋯ |
| L(s) = 1 | + (−0.425 − 1.31i)2-s + (−0.726 + 0.527i)4-s + (0.298 + 0.954i)5-s + 0.339·7-s + (−0.114 − 0.0829i)8-s + (1.12 − 0.797i)10-s + (−0.373 − 1.15i)11-s + (−0.282 + 0.870i)13-s + (−0.144 − 0.444i)14-s + (−0.337 + 1.03i)16-s + (−1.16 − 0.846i)17-s + (−0.786 − 0.571i)19-s + (−0.720 − 0.535i)20-s + (−1.34 + 0.979i)22-s + (−0.440 − 1.35i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.620 - 0.783i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.620 - 0.783i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.149883 + 0.309883i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.149883 + 0.309883i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + (-3.34 - 10.6i)T \) |
| good | 2 | \( 1 + (1.20 + 3.70i)T + (-6.47 + 4.70i)T^{2} \) |
| 7 | \( 1 - 6.27T + 343T^{2} \) |
| 11 | \( 1 + (13.6 + 41.9i)T + (-1.07e3 + 782. i)T^{2} \) |
| 13 | \( 1 + (13.2 - 40.7i)T + (-1.77e3 - 1.29e3i)T^{2} \) |
| 17 | \( 1 + (81.6 + 59.3i)T + (1.51e3 + 4.67e3i)T^{2} \) |
| 19 | \( 1 + (65.1 + 47.3i)T + (2.11e3 + 6.52e3i)T^{2} \) |
| 23 | \( 1 + (48.5 + 149. i)T + (-9.84e3 + 7.15e3i)T^{2} \) |
| 29 | \( 1 + (199. - 145. i)T + (7.53e3 - 2.31e4i)T^{2} \) |
| 31 | \( 1 + (-69.6 - 50.6i)T + (9.20e3 + 2.83e4i)T^{2} \) |
| 37 | \( 1 + (-61.0 + 188. i)T + (-4.09e4 - 2.97e4i)T^{2} \) |
| 41 | \( 1 + (-17.2 + 53.1i)T + (-5.57e4 - 4.05e4i)T^{2} \) |
| 43 | \( 1 - 211.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (-85.9 + 62.4i)T + (3.20e4 - 9.87e4i)T^{2} \) |
| 53 | \( 1 + (153. - 111. i)T + (4.60e4 - 1.41e5i)T^{2} \) |
| 59 | \( 1 + (140. - 432. i)T + (-1.66e5 - 1.20e5i)T^{2} \) |
| 61 | \( 1 + (-171. - 528. i)T + (-1.83e5 + 1.33e5i)T^{2} \) |
| 67 | \( 1 + (773. + 561. i)T + (9.29e4 + 2.86e5i)T^{2} \) |
| 71 | \( 1 + (-454. + 330. i)T + (1.10e5 - 3.40e5i)T^{2} \) |
| 73 | \( 1 + (54.7 + 168. i)T + (-3.14e5 + 2.28e5i)T^{2} \) |
| 79 | \( 1 + (769. - 558. i)T + (1.52e5 - 4.68e5i)T^{2} \) |
| 83 | \( 1 + (-11.3 - 8.23i)T + (1.76e5 + 5.43e5i)T^{2} \) |
| 89 | \( 1 + (143. + 440. i)T + (-5.70e5 + 4.14e5i)T^{2} \) |
| 97 | \( 1 + (-1.05e3 + 764. i)T + (2.82e5 - 8.68e5i)T^{2} \) |
| 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
−10.95569157770816704611119275738, −10.64987766713945145931582492957, −9.352846646001680655912685683467, −8.662830384321041483571401466747, −7.09270992710745472257960212797, −6.08966618799291643637321052546, −4.33255443362769863010070844765, −2.91801937385713314027686635632, −2.06419912976957575392003709363, −0.14996463020033908168475125782,
1.98953822201703979465081796429, 4.36829528376315674317632899688, 5.41610515333012203408803140869, 6.31024890529096174247634470329, 7.68953103038469668422808840271, 8.154571821335482287187004284755, 9.271171949453616768629540205708, 10.06193808525621511002714474955, 11.50516840660108787315439553964, 12.66798654445565229810405909372
