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
−12.66007662057760968651666297573, −12.00302871405255403126013572042, −11.11806449642176787545309767895, −9.613193216268349878769413572098, −8.101848446483000836794877889196, −6.76243760149713490457385835169, −6.26283957023794563254966422390, −4.44534276551543698073604456770, −2.16684064658485241684613150971, −0.981135889361308549780183988939,
0.863851999984749370434135308312, 3.30633643316194640521711616050, 4.84823341883251469714100246465, 5.48625465916613203763122614869, 7.26297373540407154898452001537, 9.046100551067609950708434874074, 9.802432157410888492614915052250, 10.96319506879720151817201109934, 11.77865906360317064495462498651, 13.06422161062906924999741526172