| L(s) = 1 | + (−0.266 − 0.333i)2-s + (−0.260 − 1.71i)3-s + (0.404 − 1.77i)4-s + (−1.62 + 0.781i)5-s + (−0.502 + 0.542i)6-s − 1.03i·7-s + (−1.46 + 0.707i)8-s + (−2.86 + 0.891i)9-s + (0.693 + 0.333i)10-s + (1.08 − 0.246i)11-s + (−3.13 − 0.231i)12-s + (2.87 − 1.38i)13-s + (−0.344 + 0.274i)14-s + (1.76 + 2.57i)15-s + (−2.64 − 1.27i)16-s + (1.67 − 3.48i)17-s + ⋯ |
| L(s) = 1 | + (−0.188 − 0.235i)2-s + (−0.150 − 0.988i)3-s + (0.202 − 0.886i)4-s + (−0.726 + 0.349i)5-s + (−0.205 + 0.221i)6-s − 0.389i·7-s + (−0.519 + 0.250i)8-s + (−0.954 + 0.297i)9-s + (0.219 + 0.105i)10-s + (0.325 − 0.0744i)11-s + (−0.906 − 0.0667i)12-s + (0.798 − 0.384i)13-s + (−0.0919 + 0.0733i)14-s + (0.454 + 0.665i)15-s + (−0.662 − 0.318i)16-s + (0.406 − 0.844i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 129 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.500 + 0.865i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 129 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.500 + 0.865i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.422119 - 0.731985i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.422119 - 0.731985i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.260 + 1.71i)T \) |
| 43 | \( 1 + (-4.59 - 4.67i)T \) |
| good | 2 | \( 1 + (0.266 + 0.333i)T + (-0.445 + 1.94i)T^{2} \) |
| 5 | \( 1 + (1.62 - 0.781i)T + (3.11 - 3.90i)T^{2} \) |
| 7 | \( 1 + 1.03iT - 7T^{2} \) |
| 11 | \( 1 + (-1.08 + 0.246i)T + (9.91 - 4.77i)T^{2} \) |
| 13 | \( 1 + (-2.87 + 1.38i)T + (8.10 - 10.1i)T^{2} \) |
| 17 | \( 1 + (-1.67 + 3.48i)T + (-10.5 - 13.2i)T^{2} \) |
| 19 | \( 1 + (-2.99 - 0.684i)T + (17.1 + 8.24i)T^{2} \) |
| 23 | \( 1 + (-7.24 + 1.65i)T + (20.7 - 9.97i)T^{2} \) |
| 29 | \( 1 + (1.20 + 1.50i)T + (-6.45 + 28.2i)T^{2} \) |
| 31 | \( 1 + (-1.30 - 1.63i)T + (-6.89 + 30.2i)T^{2} \) |
| 37 | \( 1 - 3.68iT - 37T^{2} \) |
| 41 | \( 1 + (6.73 - 5.36i)T + (9.12 - 39.9i)T^{2} \) |
| 47 | \( 1 + (4.92 + 1.12i)T + (42.3 + 20.3i)T^{2} \) |
| 53 | \( 1 + (-5.74 + 11.9i)T + (-33.0 - 41.4i)T^{2} \) |
| 59 | \( 1 + (-2.55 + 5.30i)T + (-36.7 - 46.1i)T^{2} \) |
| 61 | \( 1 + (8.53 + 6.80i)T + (13.5 + 59.4i)T^{2} \) |
| 67 | \( 1 + (1.48 - 6.49i)T + (-60.3 - 29.0i)T^{2} \) |
| 71 | \( 1 + (3.67 - 16.1i)T + (-63.9 - 30.8i)T^{2} \) |
| 73 | \( 1 + (-0.351 - 0.729i)T + (-45.5 + 57.0i)T^{2} \) |
| 79 | \( 1 + 10.0T + 79T^{2} \) |
| 83 | \( 1 + (0.0870 + 0.0694i)T + (18.4 + 80.9i)T^{2} \) |
| 89 | \( 1 + (5.63 - 7.06i)T + (-19.8 - 86.7i)T^{2} \) |
| 97 | \( 1 + (3.14 + 13.7i)T + (-87.3 + 42.0i)T^{2} \) |
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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
−13.00819980685745625772904204764, −11.53794286847605317086475319336, −11.33408137690825603802897557940, −10.03085283323725423333038533012, −8.664653712528222090393621483896, −7.41965381057813531696010506850, −6.53749162098457478723061295303, −5.25594029890992390033261471478, −3.10827348918178111544387777278, −1.07099110837180886321820785536,
3.30192158395247942756279959856, 4.29019445871126109588934710497, 5.85969537426514093600395161213, 7.35992015538056579196560137788, 8.614076280552757452768468410143, 9.132946795114203754989980042466, 10.70765216162716369544194920616, 11.69202538786553686452841134238, 12.32322613240289049143760736878, 13.67394192303796460000985219129
