| L(s) = 1 | + (−2.35 − 3.23i)2-s + (17.7 + 5.76i)3-s + (−4.94 + 15.2i)4-s + (−50.3 − 24.2i)5-s + (−23.0 − 71.0i)6-s − 135. i·7-s + (60.8 − 19.7i)8-s + (85.2 + 61.9i)9-s + (39.8 + 220. i)10-s + (607. − 441. i)11-s + (−175. + 241. i)12-s + (147. − 202. i)13-s + (−438. + 318. i)14-s + (−753. − 721. i)15-s + (−207. − 150. i)16-s + (−533. + 173. i)17-s + ⋯ |
| L(s) = 1 | + (−0.415 − 0.572i)2-s + (1.13 + 0.370i)3-s + (−0.154 + 0.475i)4-s + (−0.900 − 0.434i)5-s + (−0.261 − 0.805i)6-s − 1.04i·7-s + (0.336 − 0.109i)8-s + (0.350 + 0.254i)9-s + (0.126 + 0.695i)10-s + (1.51 − 1.10i)11-s + (−0.351 + 0.484i)12-s + (0.241 − 0.332i)13-s + (−0.597 + 0.434i)14-s + (−0.865 − 0.827i)15-s + (−0.202 − 0.146i)16-s + (−0.447 + 0.145i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.143 + 0.989i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.143 + 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.01769 - 1.17599i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.01769 - 1.17599i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (2.35 + 3.23i)T \) |
| 5 | \( 1 + (50.3 + 24.2i)T \) |
| good | 3 | \( 1 + (-17.7 - 5.76i)T + (196. + 142. i)T^{2} \) |
| 7 | \( 1 + 135. iT - 1.68e4T^{2} \) |
| 11 | \( 1 + (-607. + 441. i)T + (4.97e4 - 1.53e5i)T^{2} \) |
| 13 | \( 1 + (-147. + 202. i)T + (-1.14e5 - 3.53e5i)T^{2} \) |
| 17 | \( 1 + (533. - 173. i)T + (1.14e6 - 8.34e5i)T^{2} \) |
| 19 | \( 1 + (713. + 2.19e3i)T + (-2.00e6 + 1.45e6i)T^{2} \) |
| 23 | \( 1 + (-486. - 669. i)T + (-1.98e6 + 6.12e6i)T^{2} \) |
| 29 | \( 1 + (1.57e3 - 4.84e3i)T + (-1.65e7 - 1.20e7i)T^{2} \) |
| 31 | \( 1 + (-1.13e3 - 3.50e3i)T + (-2.31e7 + 1.68e7i)T^{2} \) |
| 37 | \( 1 + (2.96e3 - 4.08e3i)T + (-2.14e7 - 6.59e7i)T^{2} \) |
| 41 | \( 1 + (-9.19e3 - 6.67e3i)T + (3.58e7 + 1.10e8i)T^{2} \) |
| 43 | \( 1 + 2.03e4iT - 1.47e8T^{2} \) |
| 47 | \( 1 + (-1.48e3 - 483. i)T + (1.85e8 + 1.34e8i)T^{2} \) |
| 53 | \( 1 + (-6.30e3 - 2.04e3i)T + (3.38e8 + 2.45e8i)T^{2} \) |
| 59 | \( 1 + (-3.66e4 - 2.66e4i)T + (2.20e8 + 6.79e8i)T^{2} \) |
| 61 | \( 1 + (3.98e4 - 2.89e4i)T + (2.60e8 - 8.03e8i)T^{2} \) |
| 67 | \( 1 + (-5.18e4 + 1.68e4i)T + (1.09e9 - 7.93e8i)T^{2} \) |
| 71 | \( 1 + (1.37e4 - 4.21e4i)T + (-1.45e9 - 1.06e9i)T^{2} \) |
| 73 | \( 1 + (1.31e4 + 1.80e4i)T + (-6.40e8 + 1.97e9i)T^{2} \) |
| 79 | \( 1 + (-1.29e4 + 3.98e4i)T + (-2.48e9 - 1.80e9i)T^{2} \) |
| 83 | \( 1 + (2.36e4 - 7.69e3i)T + (3.18e9 - 2.31e9i)T^{2} \) |
| 89 | \( 1 + (5.60e3 - 4.07e3i)T + (1.72e9 - 5.31e9i)T^{2} \) |
| 97 | \( 1 + (-2.81e3 - 914. i)T + (6.94e9 + 5.04e9i)T^{2} \) |
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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
−14.13191276922236919214637191975, −13.21955212591187476271834399871, −11.67451707633492797225519731135, −10.70794196936416669033882627574, −9.027767104010966937958747095078, −8.591072853238414798859451001867, −7.10743179382047750499323771843, −4.17758479478689021243134363044, −3.31507395860133933791248571376, −0.857229712898148101745694491609,
2.08715026588070819079165325106, 4.02454568265609912672849381364, 6.38394986636600286914708882217, 7.63365521005156440864587951217, 8.628978654843935521400735325837, 9.561443984916857593396366081424, 11.46885229436403772699703243288, 12.57679335951577840759702877242, 14.30067478836514211347791428397, 14.77888455398660299449658939193
