| L(s) = 1 | + (−0.752 − 1.19i)2-s + (1.71 − 4.89i)3-s + (−0.867 + 1.80i)4-s + (1.85 + 0.423i)5-s + (−7.15 + 1.63i)6-s + (−4.77 + 2.29i)7-s + (2.81 − 0.316i)8-s + (−13.9 − 11.1i)9-s + (−0.889 − 2.54i)10-s + (13.0 + 1.47i)11-s + (7.33 + 7.33i)12-s + (2.97 − 2.37i)13-s + (6.34 + 3.98i)14-s + (5.25 − 8.35i)15-s + (−2.49 − 3.12i)16-s + (−14.4 + 14.4i)17-s + ⋯ |
| L(s) = 1 | + (−0.376 − 0.598i)2-s + (0.570 − 1.63i)3-s + (−0.216 + 0.450i)4-s + (0.371 + 0.0847i)5-s + (−1.19 + 0.272i)6-s + (−0.681 + 0.328i)7-s + (0.351 − 0.0395i)8-s + (−1.55 − 1.24i)9-s + (−0.0889 − 0.254i)10-s + (1.18 + 0.133i)11-s + (0.611 + 0.611i)12-s + (0.228 − 0.182i)13-s + (0.452 + 0.284i)14-s + (0.350 − 0.557i)15-s + (−0.155 − 0.195i)16-s + (−0.848 + 0.848i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 58 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.393 + 0.919i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 58 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.393 + 0.919i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.629230 - 0.953587i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.629230 - 0.953587i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.752 + 1.19i)T \) |
| 29 | \( 1 + (-28.9 + 1.88i)T \) |
| good | 3 | \( 1 + (-1.71 + 4.89i)T + (-7.03 - 5.61i)T^{2} \) |
| 5 | \( 1 + (-1.85 - 0.423i)T + (22.5 + 10.8i)T^{2} \) |
| 7 | \( 1 + (4.77 - 2.29i)T + (30.5 - 38.3i)T^{2} \) |
| 11 | \( 1 + (-13.0 - 1.47i)T + (117. + 26.9i)T^{2} \) |
| 13 | \( 1 + (-2.97 + 2.37i)T + (37.6 - 164. i)T^{2} \) |
| 17 | \( 1 + (14.4 - 14.4i)T - 289iT^{2} \) |
| 19 | \( 1 + (-34.6 + 12.1i)T + (282. - 225. i)T^{2} \) |
| 23 | \( 1 + (-6.87 - 30.1i)T + (-476. + 229. i)T^{2} \) |
| 31 | \( 1 + (0.355 + 0.565i)T + (-416. + 865. i)T^{2} \) |
| 37 | \( 1 + (27.7 - 3.12i)T + (1.33e3 - 304. i)T^{2} \) |
| 41 | \( 1 + (13.8 + 13.8i)T + 1.68e3iT^{2} \) |
| 43 | \( 1 + (52.2 + 32.8i)T + (802. + 1.66e3i)T^{2} \) |
| 47 | \( 1 + (9.82 - 87.2i)T + (-2.15e3 - 491. i)T^{2} \) |
| 53 | \( 1 + (5.60 - 24.5i)T + (-2.53e3 - 1.21e3i)T^{2} \) |
| 59 | \( 1 + 28.3T + 3.48e3T^{2} \) |
| 61 | \( 1 + (12.3 - 35.4i)T + (-2.90e3 - 2.32e3i)T^{2} \) |
| 67 | \( 1 + (-28.8 - 23.0i)T + (998. + 4.37e3i)T^{2} \) |
| 71 | \( 1 + (-42.8 + 34.1i)T + (1.12e3 - 4.91e3i)T^{2} \) |
| 73 | \( 1 + (22.4 - 35.7i)T + (-2.31e3 - 4.80e3i)T^{2} \) |
| 79 | \( 1 + (10.9 + 97.5i)T + (-6.08e3 + 1.38e3i)T^{2} \) |
| 83 | \( 1 + (41.1 + 19.8i)T + (4.29e3 + 5.38e3i)T^{2} \) |
| 89 | \( 1 + (27.9 + 44.5i)T + (-3.43e3 + 7.13e3i)T^{2} \) |
| 97 | \( 1 + (-6.70 - 19.1i)T + (-7.35e3 + 5.86e3i)T^{2} \) |
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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
−13.99727750011009755551832444035, −13.41603702166880339284902073383, −12.34375239245615893027951130531, −11.52961533171199229142497281085, −9.621448684020169309158588242394, −8.691930955859863770629634888422, −7.31207169134082250234555686821, −6.21992184239694263296694847116, −3.16087910927431389045574672517, −1.51604029121373908519865259173,
3.53024818665357486868803143930, 4.99367453576886174680562915872, 6.65723540169099987479683162510, 8.559861707513594564917359932746, 9.506015056919979682487791105953, 10.08493062820400785680913832256, 11.53528612796658723006323609268, 13.68993786925986729416604736269, 14.32319391356583104932609745420, 15.49851072910201745695790068863
