| L(s) = 1 | + (−5.64 + 0.364i)2-s + (2.01 − 3.48i)3-s + (31.7 − 4.11i)4-s + (54.6 − 94.7i)5-s + (−10.0 + 20.4i)6-s − 177. i·7-s + (−177. + 34.8i)8-s + (113. + 196. i)9-s + (−274. + 554. i)10-s + 61.3i·11-s + (49.5 − 118. i)12-s + (252. − 145. i)13-s + (64.5 + 9.99e2i)14-s + (−220. − 381. i)15-s + (990. − 261. i)16-s + (269. − 467. i)17-s + ⋯ |
| L(s) = 1 | + (−0.997 + 0.0644i)2-s + (0.129 − 0.223i)3-s + (0.991 − 0.128i)4-s + (0.978 − 1.69i)5-s + (−0.114 + 0.231i)6-s − 1.36i·7-s + (−0.981 + 0.192i)8-s + (0.466 + 0.808i)9-s + (−0.866 + 1.75i)10-s + 0.152i·11-s + (0.0993 − 0.238i)12-s + (0.414 − 0.239i)13-s + (0.0880 + 1.36i)14-s + (−0.252 − 0.437i)15-s + (0.966 − 0.255i)16-s + (0.226 − 0.392i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.443 + 0.896i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.443 + 0.896i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.700493 - 1.12770i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.700493 - 1.12770i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (5.64 - 0.364i)T \) |
| 19 | \( 1 + (1.47e3 - 556. i)T \) |
| good | 3 | \( 1 + (-2.01 + 3.48i)T + (-121.5 - 210. i)T^{2} \) |
| 5 | \( 1 + (-54.6 + 94.7i)T + (-1.56e3 - 2.70e3i)T^{2} \) |
| 7 | \( 1 + 177. iT - 1.68e4T^{2} \) |
| 11 | \( 1 - 61.3iT - 1.61e5T^{2} \) |
| 13 | \( 1 + (-252. + 145. i)T + (1.85e5 - 3.21e5i)T^{2} \) |
| 17 | \( 1 + (-269. + 467. i)T + (-7.09e5 - 1.22e6i)T^{2} \) |
| 23 | \( 1 + (2.54e3 - 1.46e3i)T + (3.21e6 - 5.57e6i)T^{2} \) |
| 29 | \( 1 + (-3.74e3 + 2.16e3i)T + (1.02e7 - 1.77e7i)T^{2} \) |
| 31 | \( 1 - 5.97e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 2.10e3iT - 6.93e7T^{2} \) |
| 41 | \( 1 + (-8.33e3 - 4.81e3i)T + (5.79e7 + 1.00e8i)T^{2} \) |
| 43 | \( 1 + (1.62e4 + 9.35e3i)T + (7.35e7 + 1.27e8i)T^{2} \) |
| 47 | \( 1 + (-1.38e3 + 802. i)T + (1.14e8 - 1.98e8i)T^{2} \) |
| 53 | \( 1 + (1.73e4 - 1.00e4i)T + (2.09e8 - 3.62e8i)T^{2} \) |
| 59 | \( 1 + (1.75e4 - 3.04e4i)T + (-3.57e8 - 6.19e8i)T^{2} \) |
| 61 | \( 1 + (-6.46e3 - 1.12e4i)T + (-4.22e8 + 7.31e8i)T^{2} \) |
| 67 | \( 1 + (2.56e3 + 4.44e3i)T + (-6.75e8 + 1.16e9i)T^{2} \) |
| 71 | \( 1 + (1.96e4 - 3.40e4i)T + (-9.02e8 - 1.56e9i)T^{2} \) |
| 73 | \( 1 + (-2.55e4 + 4.42e4i)T + (-1.03e9 - 1.79e9i)T^{2} \) |
| 79 | \( 1 + (-4.35e4 + 7.53e4i)T + (-1.53e9 - 2.66e9i)T^{2} \) |
| 83 | \( 1 + 3.29e4iT - 3.93e9T^{2} \) |
| 89 | \( 1 + (4.14e4 - 2.39e4i)T + (2.79e9 - 4.83e9i)T^{2} \) |
| 97 | \( 1 + (-7.77e4 - 4.48e4i)T + (4.29e9 + 7.43e9i)T^{2} \) |
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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
−13.25451622997817308819417280226, −12.12515902915943667831157979327, −10.43999731332705939763841564252, −9.847138084496790035037645891729, −8.525754081852627573167063355504, −7.67836859502626097455264129093, −6.12311594813767336559102223521, −4.55789918879537439319295056372, −1.82647769705646910741316277981, −0.793258757442140814699668980834,
2.03076061249507322982952782838, 3.12479879831694866772414629581, 6.16352983024267528854717499020, 6.58689507976279701817536581656, 8.396883114499804250197437028153, 9.541282966153964197105933210128, 10.29881642798533508799115285866, 11.34733430313153091839120695832, 12.52023758466668350790375171838, 14.25937974758519204204482760907
