| L(s) = 1 | + (32.2 − 11.7i)3-s + (−40.5 + 230. i)5-s + (192. + 333. i)7-s + (−773. + 648. i)9-s + (−15.6 + 27.0i)11-s + (−8.53e3 − 3.10e3i)13-s + (1.39e3 + 7.89e3i)15-s + (−1.12e4 − 9.46e3i)17-s + (−2.69e4 + 1.30e4i)19-s + (1.01e4 + 8.50e3i)21-s + (−1.87e3 − 1.06e4i)23-s + (2.21e4 + 8.04e3i)25-s + (−5.48e4 + 9.49e4i)27-s + (−5.70e3 + 4.78e3i)29-s + (3.94e4 + 6.83e4i)31-s + ⋯ |
| L(s) = 1 | + (0.689 − 0.250i)3-s + (−0.145 + 0.823i)5-s + (0.212 + 0.367i)7-s + (−0.353 + 0.296i)9-s + (−0.00353 + 0.00612i)11-s + (−1.07 − 0.392i)13-s + (0.106 + 0.604i)15-s + (−0.556 − 0.467i)17-s + (−0.899 + 0.436i)19-s + (0.238 + 0.200i)21-s + (−0.0321 − 0.182i)23-s + (0.282 + 0.102i)25-s + (−0.536 + 0.928i)27-s + (−0.0434 + 0.0364i)29-s + (0.237 + 0.412i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.836 - 0.548i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.836 - 0.548i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.275045 + 0.920965i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.275045 + 0.920965i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + (2.69e4 - 1.30e4i)T \) |
| good | 3 | \( 1 + (-32.2 + 11.7i)T + (1.67e3 - 1.40e3i)T^{2} \) |
| 5 | \( 1 + (40.5 - 230. i)T + (-7.34e4 - 2.67e4i)T^{2} \) |
| 7 | \( 1 + (-192. - 333. i)T + (-4.11e5 + 7.13e5i)T^{2} \) |
| 11 | \( 1 + (15.6 - 27.0i)T + (-9.74e6 - 1.68e7i)T^{2} \) |
| 13 | \( 1 + (8.53e3 + 3.10e3i)T + (4.80e7 + 4.03e7i)T^{2} \) |
| 17 | \( 1 + (1.12e4 + 9.46e3i)T + (7.12e7 + 4.04e8i)T^{2} \) |
| 23 | \( 1 + (1.87e3 + 1.06e4i)T + (-3.19e9 + 1.16e9i)T^{2} \) |
| 29 | \( 1 + (5.70e3 - 4.78e3i)T + (2.99e9 - 1.69e10i)T^{2} \) |
| 31 | \( 1 + (-3.94e4 - 6.83e4i)T + (-1.37e10 + 2.38e10i)T^{2} \) |
| 37 | \( 1 + 3.43e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + (5.28e5 - 1.92e5i)T + (1.49e11 - 1.25e11i)T^{2} \) |
| 43 | \( 1 + (1.16e5 - 6.60e5i)T + (-2.55e11 - 9.29e10i)T^{2} \) |
| 47 | \( 1 + (5.03e5 - 4.22e5i)T + (8.79e10 - 4.98e11i)T^{2} \) |
| 53 | \( 1 + (-1.26e5 - 7.20e5i)T + (-1.10e12 + 4.01e11i)T^{2} \) |
| 59 | \( 1 + (-8.98e5 - 7.53e5i)T + (4.32e11 + 2.45e12i)T^{2} \) |
| 61 | \( 1 + (-6.84e4 - 3.88e5i)T + (-2.95e12 + 1.07e12i)T^{2} \) |
| 67 | \( 1 + (-2.19e6 + 1.83e6i)T + (1.05e12 - 5.96e12i)T^{2} \) |
| 71 | \( 1 + (-3.13e5 + 1.77e6i)T + (-8.54e12 - 3.11e12i)T^{2} \) |
| 73 | \( 1 + (-1.11e6 + 4.07e5i)T + (8.46e12 - 7.10e12i)T^{2} \) |
| 79 | \( 1 + (-3.33e6 + 1.21e6i)T + (1.47e13 - 1.23e13i)T^{2} \) |
| 83 | \( 1 + (3.49e5 + 6.05e5i)T + (-1.35e13 + 2.35e13i)T^{2} \) |
| 89 | \( 1 + (-2.23e6 - 8.12e5i)T + (3.38e13 + 2.84e13i)T^{2} \) |
| 97 | \( 1 + (-8.52e6 - 7.15e6i)T + (1.40e13 + 7.95e13i)T^{2} \) |
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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
−13.68103321296727709211720516367, −12.47838830431658896409393419208, −11.27915152316586587596329736517, −10.20967276499175043982913016793, −8.814395470162482149517732225833, −7.79852263921811802072910795319, −6.66113513253617544155137272326, −5.00763370144759257336230338221, −3.13519588349137754442606228622, −2.13781559150813278651125926607,
0.27256101806726428728093654929, 2.16207260954469560426569076532, 3.85993691580605008771445759430, 5.03118097585861672905994241529, 6.81447490584599366618316361379, 8.300894041301168285423268489153, 9.008151984853676378223204383424, 10.21092596914880237131202473314, 11.63365881269687205663542338867, 12.67361745798546970337644482636
