| L(s) = 1 | + (−0.764 + 1.18i)2-s + (−0.830 − 1.81i)4-s + (−2.17 + 7.41i)5-s + (6.85 − 7.91i)7-s + (2.79 + 0.402i)8-s + (−7.15 − 8.25i)10-s + (−0.214 − 0.333i)11-s + (7.09 + 8.18i)13-s + (4.17 + 14.2i)14-s + (−2.61 + 3.02i)16-s + (9.43 + 4.31i)17-s + (6.43 + 14.0i)19-s + (15.2 − 2.19i)20-s + 0.561·22-s + (21.1 + 9.03i)23-s + ⋯ |
| L(s) = 1 | + (−0.382 + 0.594i)2-s + (−0.207 − 0.454i)4-s + (−0.435 + 1.48i)5-s + (0.979 − 1.13i)7-s + (0.349 + 0.0503i)8-s + (−0.715 − 0.825i)10-s + (−0.0195 − 0.0303i)11-s + (0.545 + 0.629i)13-s + (0.297 + 1.01i)14-s + (−0.163 + 0.188i)16-s + (0.555 + 0.253i)17-s + (0.338 + 0.742i)19-s + (0.764 − 0.109i)20-s + 0.0255·22-s + (0.919 + 0.392i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.385 - 0.922i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.385 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.733050 + 1.10011i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.733050 + 1.10011i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.764 - 1.18i)T \) |
| 3 | \( 1 \) |
| 23 | \( 1 + (-21.1 - 9.03i)T \) |
| good | 5 | \( 1 + (2.17 - 7.41i)T + (-21.0 - 13.5i)T^{2} \) |
| 7 | \( 1 + (-6.85 + 7.91i)T + (-6.97 - 48.5i)T^{2} \) |
| 11 | \( 1 + (0.214 + 0.333i)T + (-50.2 + 110. i)T^{2} \) |
| 13 | \( 1 + (-7.09 - 8.18i)T + (-24.0 + 167. i)T^{2} \) |
| 17 | \( 1 + (-9.43 - 4.31i)T + (189. + 218. i)T^{2} \) |
| 19 | \( 1 + (-6.43 - 14.0i)T + (-236. + 272. i)T^{2} \) |
| 29 | \( 1 + (38.3 + 17.5i)T + (550. + 635. i)T^{2} \) |
| 31 | \( 1 + (1.84 - 12.8i)T + (-922. - 270. i)T^{2} \) |
| 37 | \( 1 + (25.4 - 7.47i)T + (1.15e3 - 740. i)T^{2} \) |
| 41 | \( 1 + (14.1 - 48.3i)T + (-1.41e3 - 908. i)T^{2} \) |
| 43 | \( 1 + (-2.60 - 18.0i)T + (-1.77e3 + 520. i)T^{2} \) |
| 47 | \( 1 - 47.1iT - 2.20e3T^{2} \) |
| 53 | \( 1 + (-40.3 - 34.9i)T + (399. + 2.78e3i)T^{2} \) |
| 59 | \( 1 + (36.5 - 31.6i)T + (495. - 3.44e3i)T^{2} \) |
| 61 | \( 1 + (9.80 - 68.2i)T + (-3.57e3 - 1.04e3i)T^{2} \) |
| 67 | \( 1 + (-79.4 - 51.0i)T + (1.86e3 + 4.08e3i)T^{2} \) |
| 71 | \( 1 + (-59.1 + 92.0i)T + (-2.09e3 - 4.58e3i)T^{2} \) |
| 73 | \( 1 + (6.28 + 13.7i)T + (-3.48e3 + 4.02e3i)T^{2} \) |
| 79 | \( 1 + (2.59 + 2.99i)T + (-888. + 6.17e3i)T^{2} \) |
| 83 | \( 1 + (29.9 + 101. i)T + (-5.79e3 + 3.72e3i)T^{2} \) |
| 89 | \( 1 + (96.4 - 13.8i)T + (7.60e3 - 2.23e3i)T^{2} \) |
| 97 | \( 1 + (-136. - 40.0i)T + (7.91e3 + 5.08e3i)T^{2} \) |
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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
−11.01408723519302294232935558306, −10.55627074831534905853177236691, −9.545609544891197977610836262930, −8.196265234145778942001428216036, −7.48837003783233515856954524437, −6.91493999438872635075905563368, −5.80574762152491738057436147758, −4.36556716201513109116226802775, −3.35519280826466853529813745140, −1.44331166992184313001883479839,
0.72763336013003031090090608384, 2.01587504848794587877227084305, 3.60448710607715820092140644942, 5.02521452936806296093617154431, 5.40408052681787463833671875725, 7.35163279730058525327583834904, 8.460746381063545394847195175519, 8.700235974281761675330988871678, 9.605349776601152872438839182890, 10.98664231582871802594123479611
