| L(s) = 1 | + (1.31 − 4.04i)2-s + (−2.42 − 1.76i)3-s + (−8.13 − 5.91i)4-s + (−8.20 + 7.59i)5-s + (−10.3 + 7.49i)6-s − 28.2·7-s + (−7.06 + 5.13i)8-s + (2.78 + 8.55i)9-s + (19.8 + 43.1i)10-s + (15.8 − 48.6i)11-s + (9.32 + 28.6i)12-s + (−7.92 − 24.4i)13-s + (−37.1 + 114. i)14-s + (33.3 − 3.94i)15-s + (−13.3 − 41.2i)16-s + (−75.4 + 54.8i)17-s + ⋯ |
| L(s) = 1 | + (0.464 − 1.42i)2-s + (−0.467 − 0.339i)3-s + (−1.01 − 0.738i)4-s + (−0.734 + 0.678i)5-s + (−0.701 + 0.509i)6-s − 1.52·7-s + (−0.312 + 0.226i)8-s + (0.103 + 0.317i)9-s + (0.629 + 1.36i)10-s + (0.433 − 1.33i)11-s + (0.224 + 0.690i)12-s + (−0.169 − 0.520i)13-s + (−0.709 + 2.18i)14-s + (0.573 − 0.0679i)15-s + (−0.209 − 0.643i)16-s + (−1.07 + 0.782i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.718 - 0.695i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.718 - 0.695i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.261637 + 0.646379i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.261637 + 0.646379i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (2.42 + 1.76i)T \) |
| 5 | \( 1 + (8.20 - 7.59i)T \) |
| good | 2 | \( 1 + (-1.31 + 4.04i)T + (-6.47 - 4.70i)T^{2} \) |
| 7 | \( 1 + 28.2T + 343T^{2} \) |
| 11 | \( 1 + (-15.8 + 48.6i)T + (-1.07e3 - 782. i)T^{2} \) |
| 13 | \( 1 + (7.92 + 24.4i)T + (-1.77e3 + 1.29e3i)T^{2} \) |
| 17 | \( 1 + (75.4 - 54.8i)T + (1.51e3 - 4.67e3i)T^{2} \) |
| 19 | \( 1 + (-95.3 + 69.3i)T + (2.11e3 - 6.52e3i)T^{2} \) |
| 23 | \( 1 + (29.1 - 89.8i)T + (-9.84e3 - 7.15e3i)T^{2} \) |
| 29 | \( 1 + (46.7 + 33.9i)T + (7.53e3 + 2.31e4i)T^{2} \) |
| 31 | \( 1 + (-32.7 + 23.8i)T + (9.20e3 - 2.83e4i)T^{2} \) |
| 37 | \( 1 + (126. + 389. i)T + (-4.09e4 + 2.97e4i)T^{2} \) |
| 41 | \( 1 + (-31.4 - 96.7i)T + (-5.57e4 + 4.05e4i)T^{2} \) |
| 43 | \( 1 + 1.56T + 7.95e4T^{2} \) |
| 47 | \( 1 + (-265. - 193. i)T + (3.20e4 + 9.87e4i)T^{2} \) |
| 53 | \( 1 + (-368. - 267. i)T + (4.60e4 + 1.41e5i)T^{2} \) |
| 59 | \( 1 + (120. + 369. i)T + (-1.66e5 + 1.20e5i)T^{2} \) |
| 61 | \( 1 + (149. - 459. i)T + (-1.83e5 - 1.33e5i)T^{2} \) |
| 67 | \( 1 + (122. - 88.9i)T + (9.29e4 - 2.86e5i)T^{2} \) |
| 71 | \( 1 + (158. + 115. i)T + (1.10e5 + 3.40e5i)T^{2} \) |
| 73 | \( 1 + (-186. + 573. i)T + (-3.14e5 - 2.28e5i)T^{2} \) |
| 79 | \( 1 + (484. + 352. i)T + (1.52e5 + 4.68e5i)T^{2} \) |
| 83 | \( 1 + (-654. + 475. i)T + (1.76e5 - 5.43e5i)T^{2} \) |
| 89 | \( 1 + (201. - 620. i)T + (-5.70e5 - 4.14e5i)T^{2} \) |
| 97 | \( 1 + (1.25e3 + 914. i)T + (2.82e5 + 8.68e5i)T^{2} \) |
| 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
−13.13607456024868025910374389196, −12.13796288244622726303469041553, −11.25135494978307916911671486184, −10.52396465683724012664607234762, −9.213106718542406561128641209585, −7.25972890046145550209948598580, −5.96044643893770773155960676493, −3.84937420887743082402952576873, −2.89796315136488680068037719613, −0.39980974926587139950799056507,
4.00379346040959401578306088352, 5.05554525187062499500241371515, 6.54392566633195298764848626473, 7.27300721669320095384318486473, 8.902129200596792712376124778907, 9.937296113064802866707080514412, 11.83397402363937106517567325481, 12.67714832472179521283174958568, 13.79055431963252724472806283011, 15.14205505528844869053393939203
