| L(s) = 1 | + (−0.415 − 0.909i)2-s + (1.07 − 0.314i)3-s + (−0.654 + 0.755i)4-s + (−0.915 − 0.588i)5-s + (−0.730 − 0.843i)6-s + (0.122 + 0.854i)7-s + (0.959 + 0.281i)8-s + (−1.47 + 0.949i)9-s + (−0.154 + 1.07i)10-s + (0.273 − 0.598i)11-s + (−0.463 + 1.01i)12-s + (−0.882 + 6.13i)13-s + (0.726 − 0.466i)14-s + (−1.16 − 0.341i)15-s + (−0.142 − 0.989i)16-s + (−3.72 − 4.30i)17-s + ⋯ |
| L(s) = 1 | + (−0.293 − 0.643i)2-s + (0.617 − 0.181i)3-s + (−0.327 + 0.377i)4-s + (−0.409 − 0.263i)5-s + (−0.298 − 0.344i)6-s + (0.0464 + 0.323i)7-s + (0.339 + 0.0996i)8-s + (−0.492 + 0.316i)9-s + (−0.0489 + 0.340i)10-s + (0.0823 − 0.180i)11-s + (−0.133 + 0.292i)12-s + (−0.244 + 1.70i)13-s + (0.194 − 0.124i)14-s + (−0.300 − 0.0882i)15-s + (−0.0355 − 0.247i)16-s + (−0.904 − 1.04i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 46 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.715 + 0.698i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 46 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.715 + 0.698i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.702800 - 0.286364i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.702800 - 0.286364i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.415 + 0.909i)T \) |
| 23 | \( 1 + (-3.73 + 3.01i)T \) |
| good | 3 | \( 1 + (-1.07 + 0.314i)T + (2.52 - 1.62i)T^{2} \) |
| 5 | \( 1 + (0.915 + 0.588i)T + (2.07 + 4.54i)T^{2} \) |
| 7 | \( 1 + (-0.122 - 0.854i)T + (-6.71 + 1.97i)T^{2} \) |
| 11 | \( 1 + (-0.273 + 0.598i)T + (-7.20 - 8.31i)T^{2} \) |
| 13 | \( 1 + (0.882 - 6.13i)T + (-12.4 - 3.66i)T^{2} \) |
| 17 | \( 1 + (3.72 + 4.30i)T + (-2.41 + 16.8i)T^{2} \) |
| 19 | \( 1 + (-4.22 + 4.87i)T + (-2.70 - 18.8i)T^{2} \) |
| 29 | \( 1 + (0.0667 + 0.0769i)T + (-4.12 + 28.7i)T^{2} \) |
| 31 | \( 1 + (-1.48 - 0.434i)T + (26.0 + 16.7i)T^{2} \) |
| 37 | \( 1 + (6.17 - 3.96i)T + (15.3 - 33.6i)T^{2} \) |
| 41 | \( 1 + (-5.12 - 3.29i)T + (17.0 + 37.2i)T^{2} \) |
| 43 | \( 1 + (-5.15 + 1.51i)T + (36.1 - 23.2i)T^{2} \) |
| 47 | \( 1 + 7.84T + 47T^{2} \) |
| 53 | \( 1 + (-0.676 - 4.70i)T + (-50.8 + 14.9i)T^{2} \) |
| 59 | \( 1 + (0.808 - 5.62i)T + (-56.6 - 16.6i)T^{2} \) |
| 61 | \( 1 + (0.215 + 0.0632i)T + (51.3 + 32.9i)T^{2} \) |
| 67 | \( 1 + (-0.986 - 2.15i)T + (-43.8 + 50.6i)T^{2} \) |
| 71 | \( 1 + (-2.66 - 5.84i)T + (-46.4 + 53.6i)T^{2} \) |
| 73 | \( 1 + (-2.73 + 3.15i)T + (-10.3 - 72.2i)T^{2} \) |
| 79 | \( 1 + (-2.41 + 16.8i)T + (-75.7 - 22.2i)T^{2} \) |
| 83 | \( 1 + (14.2 - 9.18i)T + (34.4 - 75.4i)T^{2} \) |
| 89 | \( 1 + (4.85 - 1.42i)T + (74.8 - 48.1i)T^{2} \) |
| 97 | \( 1 + (8.27 + 5.31i)T + (40.2 + 88.2i)T^{2} \) |
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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
−15.76907300386842336776217094409, −14.22344534019245027560178634838, −13.46936294452939021087212989803, −11.92411802570256075734584739400, −11.21617237325063970669586569126, −9.320582192335743591629558711964, −8.642985976490468625167128719492, −7.10456943922909206253801765986, −4.65812883711541595631540595394, −2.61491187203210427702432454863,
3.54842469103749240811478885644, 5.67437993554358036156859774183, 7.43551392944861718988958231567, 8.399621933552797617561003072343, 9.749690559158759410085870372693, 11.01145123788083612565222983300, 12.69601886140410914663563091794, 14.01978863047439774681977905632, 15.04652611912313646336359441705, 15.62154790269626240857992731870
