| 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 | \( 1 + (-4.51 + 5.21i)T + (-13.8 - 96.0i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.95283255589704688467988651637, −12.85334203127217393346960986743, −12.24684775942758497189570227816, −10.35179753272037217924114603917, −8.832875829275224095982878127719, −7.966070424233919088775433837452, −7.07014641748472238402665328180, −6.16342984926243006772290805739, −4.81890784170154559791292055173, −2.29389387159195127676132067673,
2.47082449458151080584000275221, 3.66764065458456539008671175641, 4.70640890673991575597517905254, 6.70276668949141836641273760151, 8.574309144029239895965413697972, 9.946679278665386220212616167772, 10.01408510354902934235095767073, 11.25248027920086218064879752605, 12.21089905565584608429578357570, 13.47131175973196362597711928381
