| L(s) = 1 | + (−34.7 + 60.1i)3-s + (55.3 − 95.9i)5-s + 314.·7-s + (−1.31e3 − 2.27e3i)9-s + 5.49e3·11-s + (−6.97e3 − 1.20e4i)13-s + (3.84e3 + 6.66e3i)15-s + (7.10e3 − 1.23e4i)17-s + (4.43e3 − 2.95e4i)19-s + (−1.09e4 + 1.88e4i)21-s + (5.50e4 + 9.52e4i)23-s + (3.29e4 + 5.70e4i)25-s + 3.09e4·27-s + (8.12e4 + 1.40e5i)29-s − 1.62e4·31-s + ⋯ |
| L(s) = 1 | + (−0.742 + 1.28i)3-s + (0.198 − 0.343i)5-s + 0.346·7-s + (−0.601 − 1.04i)9-s + 1.24·11-s + (−0.880 − 1.52i)13-s + (0.294 + 0.509i)15-s + (0.350 − 0.607i)17-s + (0.148 − 0.988i)19-s + (−0.257 + 0.445i)21-s + (0.942 + 1.63i)23-s + (0.421 + 0.729i)25-s + 0.302·27-s + (0.618 + 1.07i)29-s − 0.0979·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.832 - 0.554i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.832 - 0.554i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.57109 + 0.475286i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.57109 + 0.475286i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + (-4.43e3 + 2.95e4i)T \) |
| good | 3 | \( 1 + (34.7 - 60.1i)T + (-1.09e3 - 1.89e3i)T^{2} \) |
| 5 | \( 1 + (-55.3 + 95.9i)T + (-3.90e4 - 6.76e4i)T^{2} \) |
| 7 | \( 1 - 314.T + 8.23e5T^{2} \) |
| 11 | \( 1 - 5.49e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + (6.97e3 + 1.20e4i)T + (-3.13e7 + 5.43e7i)T^{2} \) |
| 17 | \( 1 + (-7.10e3 + 1.23e4i)T + (-2.05e8 - 3.55e8i)T^{2} \) |
| 23 | \( 1 + (-5.50e4 - 9.52e4i)T + (-1.70e9 + 2.94e9i)T^{2} \) |
| 29 | \( 1 + (-8.12e4 - 1.40e5i)T + (-8.62e9 + 1.49e10i)T^{2} \) |
| 31 | \( 1 + 1.62e4T + 2.75e10T^{2} \) |
| 37 | \( 1 - 3.09e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + (-7.82e4 + 1.35e5i)T + (-9.73e10 - 1.68e11i)T^{2} \) |
| 43 | \( 1 + (-2.60e5 + 4.50e5i)T + (-1.35e11 - 2.35e11i)T^{2} \) |
| 47 | \( 1 + (-1.37e4 - 2.37e4i)T + (-2.53e11 + 4.38e11i)T^{2} \) |
| 53 | \( 1 + (3.11e5 + 5.38e5i)T + (-5.87e11 + 1.01e12i)T^{2} \) |
| 59 | \( 1 + (-4.35e5 + 7.54e5i)T + (-1.24e12 - 2.15e12i)T^{2} \) |
| 61 | \( 1 + (-1.44e6 - 2.49e6i)T + (-1.57e12 + 2.72e12i)T^{2} \) |
| 67 | \( 1 + (-1.19e6 - 2.06e6i)T + (-3.03e12 + 5.24e12i)T^{2} \) |
| 71 | \( 1 + (-5.49e5 + 9.51e5i)T + (-4.54e12 - 7.87e12i)T^{2} \) |
| 73 | \( 1 + (2.33e5 - 4.03e5i)T + (-5.52e12 - 9.56e12i)T^{2} \) |
| 79 | \( 1 + (-3.50e6 + 6.06e6i)T + (-9.60e12 - 1.66e13i)T^{2} \) |
| 83 | \( 1 - 1.93e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + (3.27e6 + 5.66e6i)T + (-2.21e13 + 3.83e13i)T^{2} \) |
| 97 | \( 1 + (-9.65e5 + 1.67e6i)T + (-4.03e13 - 6.99e13i)T^{2} \) |
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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
−13.06422161062906924999741526172, −11.77865906360317064495462498651, −10.96319506879720151817201109934, −9.802432157410888492614915052250, −9.046100551067609950708434874074, −7.26297373540407154898452001537, −5.48625465916613203763122614869, −4.84823341883251469714100246465, −3.30633643316194640521711616050, −0.863851999984749370434135308312,
0.981135889361308549780183988939, 2.16684064658485241684613150971, 4.44534276551543698073604456770, 6.26283957023794563254966422390, 6.76243760149713490457385835169, 8.101848446483000836794877889196, 9.613193216268349878769413572098, 11.11806449642176787545309767895, 12.00302871405255403126013572042, 12.66007662057760968651666297573
