| L(s) = 1 | + (−1.01 + 3.13i)2-s + (20.3 + 14.7i)3-s + (17.0 + 12.4i)4-s + (−31.9 − 45.8i)5-s + (−67.1 + 48.7i)6-s + 70.1·7-s + (−141. + 102. i)8-s + (120. + 370. i)9-s + (176. − 53.3i)10-s + (−60.0 + 184. i)11-s + (164. + 505. i)12-s + (−295. − 909. i)13-s + (−71.5 + 220. i)14-s + (29.1 − 1.40e3i)15-s + (30.2 + 93.0i)16-s + (−71.6 + 52.0i)17-s + ⋯ |
| L(s) = 1 | + (−0.180 + 0.554i)2-s + (1.30 + 0.948i)3-s + (0.533 + 0.387i)4-s + (−0.570 − 0.821i)5-s + (−0.761 + 0.552i)6-s + 0.541·7-s + (−0.783 + 0.568i)8-s + (0.495 + 1.52i)9-s + (0.558 − 0.168i)10-s + (−0.149 + 0.460i)11-s + (0.329 + 1.01i)12-s + (−0.485 − 1.49i)13-s + (−0.0975 + 0.300i)14-s + (0.0334 − 1.61i)15-s + (0.0295 + 0.0908i)16-s + (−0.0600 + 0.0436i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.104 - 0.994i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.104 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.49565 + 1.34729i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.49565 + 1.34729i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (31.9 + 45.8i)T \) |
| good | 2 | \( 1 + (1.01 - 3.13i)T + (-25.8 - 18.8i)T^{2} \) |
| 3 | \( 1 + (-20.3 - 14.7i)T + (75.0 + 231. i)T^{2} \) |
| 7 | \( 1 - 70.1T + 1.68e4T^{2} \) |
| 11 | \( 1 + (60.0 - 184. i)T + (-1.30e5 - 9.46e4i)T^{2} \) |
| 13 | \( 1 + (295. + 909. i)T + (-3.00e5 + 2.18e5i)T^{2} \) |
| 17 | \( 1 + (71.6 - 52.0i)T + (4.38e5 - 1.35e6i)T^{2} \) |
| 19 | \( 1 + (-1.79e3 + 1.30e3i)T + (7.65e5 - 2.35e6i)T^{2} \) |
| 23 | \( 1 + (-312. + 961. i)T + (-5.20e6 - 3.78e6i)T^{2} \) |
| 29 | \( 1 + (2.42e3 + 1.76e3i)T + (6.33e6 + 1.95e7i)T^{2} \) |
| 31 | \( 1 + (-6.50e3 + 4.72e3i)T + (8.84e6 - 2.72e7i)T^{2} \) |
| 37 | \( 1 + (-4.45e3 - 1.37e4i)T + (-5.61e7 + 4.07e7i)T^{2} \) |
| 41 | \( 1 + (-886. - 2.72e3i)T + (-9.37e7 + 6.80e7i)T^{2} \) |
| 43 | \( 1 + 1.41e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + (6.88e3 + 5.00e3i)T + (7.08e7 + 2.18e8i)T^{2} \) |
| 53 | \( 1 + (3.04e4 + 2.21e4i)T + (1.29e8 + 3.97e8i)T^{2} \) |
| 59 | \( 1 + (-1.51e4 - 4.65e4i)T + (-5.78e8 + 4.20e8i)T^{2} \) |
| 61 | \( 1 + (4.51e3 - 1.39e4i)T + (-6.83e8 - 4.96e8i)T^{2} \) |
| 67 | \( 1 + (3.13e4 - 2.27e4i)T + (4.17e8 - 1.28e9i)T^{2} \) |
| 71 | \( 1 + (-3.72e3 - 2.70e3i)T + (5.57e8 + 1.71e9i)T^{2} \) |
| 73 | \( 1 + (-1.27e4 + 3.93e4i)T + (-1.67e9 - 1.21e9i)T^{2} \) |
| 79 | \( 1 + (-3.50e4 - 2.54e4i)T + (9.50e8 + 2.92e9i)T^{2} \) |
| 83 | \( 1 + (1.49e4 - 1.08e4i)T + (1.21e9 - 3.74e9i)T^{2} \) |
| 89 | \( 1 + (-5.86e3 + 1.80e4i)T + (-4.51e9 - 3.28e9i)T^{2} \) |
| 97 | \( 1 + (2.52e4 + 1.83e4i)T + (2.65e9 + 8.16e9i)T^{2} \) |
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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
−16.47441482538604498505702528047, −15.30730638722648320961713448735, −15.06754853776160118477966540345, −13.22779021216622007455985821443, −11.64214776798647259020016858846, −9.805807866303969747159678764411, −8.372224331021732936595683303605, −7.72941651127805893800040789944, −4.85910193364263178566854118736, −2.98585096943015951685605044234,
1.75867432370101131500380304918, 3.20755640127047783406021139438, 6.75059172666167358610531376280, 7.85095044402191813642621267053, 9.471269240192171383465527138593, 11.19065235331084482871890142980, 12.20121620732503066969851306416, 14.05797322681848273740687009696, 14.57645516448314315174298605824, 15.94865324171541343454189136109
