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
−12.66798654445565229810405909372, −11.50516840660108787315439553964, −10.06193808525621511002714474955, −9.271171949453616768629540205708, −8.154571821335482287187004284755, −7.68953103038469668422808840271, −6.31024890529096174247634470329, −5.41610515333012203408803140869, −4.36829528376315674317632899688, −1.98953822201703979465081796429,
0.14996463020033908168475125782, 2.06419912976957575392003709363, 2.91801937385713314027686635632, 4.33255443362769863010070844765, 6.08966618799291643637321052546, 7.09270992710745472257960212797, 8.662830384321041483571401466747, 9.352846646001680655912685683467, 10.64987766713945145931582492957, 10.95569157770816704611119275738