| L(s) = 1 | + (0.273 − 1.89i)2-s + (0.928 + 2.03i)3-s + (−1.61 − 0.474i)4-s + (0.654 + 0.755i)5-s + (4.11 − 1.20i)6-s + (−0.720 − 0.463i)7-s + (0.253 − 0.554i)8-s + (−1.30 + 1.50i)9-s + (1.61 − 1.03i)10-s + (−0.193 − 1.34i)11-s + (−0.535 − 3.72i)12-s + (−3.11 + 2.00i)13-s + (−1.07 + 1.24i)14-s + (−0.928 + 2.03i)15-s + (−3.81 − 2.45i)16-s + (−1.03 + 0.305i)17-s + ⋯ |
| L(s) = 1 | + (0.193 − 1.34i)2-s + (0.535 + 1.17i)3-s + (−0.807 − 0.237i)4-s + (0.292 + 0.337i)5-s + (1.67 − 0.493i)6-s + (−0.272 − 0.175i)7-s + (0.0894 − 0.195i)8-s + (−0.435 + 0.502i)9-s + (0.510 − 0.328i)10-s + (−0.0583 − 0.405i)11-s + (−0.154 − 1.07i)12-s + (−0.864 + 0.555i)13-s + (−0.287 + 0.332i)14-s + (−0.239 + 0.524i)15-s + (−0.953 − 0.612i)16-s + (−0.252 + 0.0740i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.760 + 0.649i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.760 + 0.649i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.24716 - 0.459921i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.24716 - 0.459921i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-0.654 - 0.755i)T \) |
| 23 | \( 1 + (4.41 - 1.86i)T \) |
| good | 2 | \( 1 + (-0.273 + 1.89i)T + (-1.91 - 0.563i)T^{2} \) |
| 3 | \( 1 + (-0.928 - 2.03i)T + (-1.96 + 2.26i)T^{2} \) |
| 7 | \( 1 + (0.720 + 0.463i)T + (2.90 + 6.36i)T^{2} \) |
| 11 | \( 1 + (0.193 + 1.34i)T + (-10.5 + 3.09i)T^{2} \) |
| 13 | \( 1 + (3.11 - 2.00i)T + (5.40 - 11.8i)T^{2} \) |
| 17 | \( 1 + (1.03 - 0.305i)T + (14.3 - 9.19i)T^{2} \) |
| 19 | \( 1 + (2.49 + 0.733i)T + (15.9 + 10.2i)T^{2} \) |
| 29 | \( 1 + (-8.83 + 2.59i)T + (24.3 - 15.6i)T^{2} \) |
| 31 | \( 1 + (-1.37 + 3.01i)T + (-20.3 - 23.4i)T^{2} \) |
| 37 | \( 1 + (6.70 - 7.74i)T + (-5.26 - 36.6i)T^{2} \) |
| 41 | \( 1 + (-6.73 - 7.77i)T + (-5.83 + 40.5i)T^{2} \) |
| 43 | \( 1 + (4.65 + 10.1i)T + (-28.1 + 32.4i)T^{2} \) |
| 47 | \( 1 + 1.15T + 47T^{2} \) |
| 53 | \( 1 + (-4.41 - 2.84i)T + (22.0 + 48.2i)T^{2} \) |
| 59 | \( 1 + (-3.37 + 2.16i)T + (24.5 - 53.6i)T^{2} \) |
| 61 | \( 1 + (-3.57 + 7.81i)T + (-39.9 - 46.1i)T^{2} \) |
| 67 | \( 1 + (0.807 - 5.61i)T + (-64.2 - 18.8i)T^{2} \) |
| 71 | \( 1 + (0.839 - 5.83i)T + (-68.1 - 20.0i)T^{2} \) |
| 73 | \( 1 + (-13.6 - 4.00i)T + (61.4 + 39.4i)T^{2} \) |
| 79 | \( 1 + (11.4 - 7.34i)T + (32.8 - 71.8i)T^{2} \) |
| 83 | \( 1 + (-5.89 + 6.80i)T + (-11.8 - 82.1i)T^{2} \) |
| 89 | \( 1 + (0.588 + 1.28i)T + (-58.2 + 67.2i)T^{2} \) |
| 97 | \( 1 + (-4.51 - 5.21i)T + (-13.8 + 96.0i)T^{2} \) |
| 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
−13.47131175973196362597711928381, −12.21089905565584608429578357570, −11.25248027920086218064879752605, −10.01408510354902934235095767073, −9.946679278665386220212616167772, −8.574309144029239895965413697972, −6.70276668949141836641273760151, −4.70640890673991575597517905254, −3.66764065458456539008671175641, −2.47082449458151080584000275221,
2.29389387159195127676132067673, 4.81890784170154559791292055173, 6.16342984926243006772290805739, 7.07014641748472238402665328180, 7.966070424233919088775433837452, 8.832875829275224095982878127719, 10.35179753272037217924114603917, 12.24684775942758497189570227816, 12.85334203127217393346960986743, 13.95283255589704688467988651637
