| L(s) = 1 | + (−0.221 − 0.435i)2-s + (0.178 − 1.12i)3-s + (1.03 − 1.42i)4-s + (−0.207 − 2.22i)5-s + (−0.529 + 0.171i)6-s + (−2.88 + 0.456i)7-s + (−1.81 − 0.287i)8-s + (1.61 + 0.525i)9-s + (−0.922 + 0.583i)10-s + (−1.41 − 1.41i)12-s + (2.60 − 1.32i)13-s + (0.837 + 1.15i)14-s + (−2.54 − 0.163i)15-s + (−0.811 − 2.49i)16-s + (−4.69 − 2.39i)17-s + (−0.130 − 0.820i)18-s + ⋯ |
| L(s) = 1 | + (−0.156 − 0.307i)2-s + (0.102 − 0.649i)3-s + (0.517 − 0.712i)4-s + (−0.0928 − 0.995i)5-s + (−0.216 + 0.0701i)6-s + (−1.08 + 0.172i)7-s + (−0.641 − 0.101i)8-s + (0.539 + 0.175i)9-s + (−0.291 + 0.184i)10-s + (−0.409 − 0.409i)12-s + (0.723 − 0.368i)13-s + (0.223 + 0.307i)14-s + (−0.656 − 0.0421i)15-s + (−0.202 − 0.624i)16-s + (−1.13 − 0.580i)17-s + (−0.0306 − 0.193i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.976 + 0.215i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.976 + 0.215i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.134305 - 1.23258i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.134305 - 1.23258i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (0.207 + 2.22i)T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + (0.221 + 0.435i)T + (-1.17 + 1.61i)T^{2} \) |
| 3 | \( 1 + (-0.178 + 1.12i)T + (-2.85 - 0.927i)T^{2} \) |
| 7 | \( 1 + (2.88 - 0.456i)T + (6.65 - 2.16i)T^{2} \) |
| 13 | \( 1 + (-2.60 + 1.32i)T + (7.64 - 10.5i)T^{2} \) |
| 17 | \( 1 + (4.69 + 2.39i)T + (9.99 + 13.7i)T^{2} \) |
| 19 | \( 1 + (-3.31 + 2.40i)T + (5.87 - 18.0i)T^{2} \) |
| 23 | \( 1 + (2.12 - 2.12i)T - 23iT^{2} \) |
| 29 | \( 1 + (-2.13 - 1.55i)T + (8.96 + 27.5i)T^{2} \) |
| 31 | \( 1 + (2.08 - 6.41i)T + (-25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (0.181 + 1.14i)T + (-35.1 + 11.4i)T^{2} \) |
| 41 | \( 1 + (2.08 + 2.87i)T + (-12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + (-5.07 - 5.07i)T + 43iT^{2} \) |
| 47 | \( 1 + (-3.63 - 0.575i)T + (44.6 + 14.5i)T^{2} \) |
| 53 | \( 1 + (0.670 + 1.31i)T + (-31.1 + 42.8i)T^{2} \) |
| 59 | \( 1 + (-0.943 + 1.29i)T + (-18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (-6.59 + 2.14i)T + (49.3 - 35.8i)T^{2} \) |
| 67 | \( 1 + (-1.31 - 1.31i)T + 67iT^{2} \) |
| 71 | \( 1 + (-0.887 - 2.73i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (2.31 + 14.6i)T + (-69.4 + 22.5i)T^{2} \) |
| 79 | \( 1 + (-4.71 + 14.4i)T + (-63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (3.03 - 5.95i)T + (-48.7 - 67.1i)T^{2} \) |
| 89 | \( 1 + 11.1iT - 89T^{2} \) |
| 97 | \( 1 + (-8.97 + 4.57i)T + (57.0 - 78.4i)T^{2} \) |
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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
−10.17207860467667887698552204273, −9.379334196267691421925496197802, −8.726841021810204088893990585678, −7.44371803696718613186932715699, −6.65966657180230847103395218981, −5.80203748731671713225383726305, −4.72714836254839140950924018366, −3.21277807225154823589697325639, −1.90256608362771677166224054878, −0.70128001105245460647347810852,
2.40491138421270824761325228502, 3.60032445328910262425092332250, 4.03140926368120679439493193812, 6.02754608807256877135081900908, 6.63510330673835814391767284171, 7.34333986880515129649801370380, 8.418257084985487253412818631449, 9.408229393033614050491283853277, 10.16269481459537680419669694645, 10.94345249720424065166368226862
