| L(s) = 1 | + (−16.8 − 9.73i)2-s + (5.24 + 9.08i)3-s + (125. + 217. i)4-s + (−269. + 155. i)5-s − 204. i·6-s + (385. + 667. i)7-s − 2.40e3i·8-s + (1.03e3 − 1.79e3i)9-s + 6.06e3·10-s + 2.03e3·11-s + (−1.31e3 + 2.28e3i)12-s + (−4.42e3 + 2.55e3i)13-s − 1.50e4i·14-s + (−2.82e3 − 1.63e3i)15-s + (−7.32e3 + 1.26e4i)16-s + (−9.54e3 − 5.50e3i)17-s + ⋯ |
| L(s) = 1 | + (−1.49 − 0.860i)2-s + (0.112 + 0.194i)3-s + (0.982 + 1.70i)4-s + (−0.964 + 0.557i)5-s − 0.386i·6-s + (0.424 + 0.735i)7-s − 1.66i·8-s + (0.474 − 0.822i)9-s + 1.91·10-s + 0.461·11-s + (−0.220 + 0.381i)12-s + (−0.558 + 0.322i)13-s − 1.46i·14-s + (−0.216 − 0.124i)15-s + (−0.446 + 0.774i)16-s + (−0.470 − 0.271i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.943 + 0.330i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.943 + 0.330i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.0347775 - 0.204259i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0347775 - 0.204259i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 37 | \( 1 + (3.31e4 + 3.06e5i)T \) |
| good | 2 | \( 1 + (16.8 + 9.73i)T + (64 + 110. i)T^{2} \) |
| 3 | \( 1 + (-5.24 - 9.08i)T + (-1.09e3 + 1.89e3i)T^{2} \) |
| 5 | \( 1 + (269. - 155. i)T + (3.90e4 - 6.76e4i)T^{2} \) |
| 7 | \( 1 + (-385. - 667. i)T + (-4.11e5 + 7.13e5i)T^{2} \) |
| 11 | \( 1 - 2.03e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + (4.42e3 - 2.55e3i)T + (3.13e7 - 5.43e7i)T^{2} \) |
| 17 | \( 1 + (9.54e3 + 5.50e3i)T + (2.05e8 + 3.55e8i)T^{2} \) |
| 19 | \( 1 + (3.39e4 - 1.96e4i)T + (4.46e8 - 7.74e8i)T^{2} \) |
| 23 | \( 1 + 3.53e4iT - 3.40e9T^{2} \) |
| 29 | \( 1 + 2.01e5iT - 1.72e10T^{2} \) |
| 31 | \( 1 - 6.82e4iT - 2.75e10T^{2} \) |
| 41 | \( 1 + (4.17e5 + 7.23e5i)T + (-9.73e10 + 1.68e11i)T^{2} \) |
| 43 | \( 1 + 1.27e5iT - 2.71e11T^{2} \) |
| 47 | \( 1 + 5.27e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + (-2.15e5 + 3.72e5i)T + (-5.87e11 - 1.01e12i)T^{2} \) |
| 59 | \( 1 + (-3.64e5 - 2.10e5i)T + (1.24e12 + 2.15e12i)T^{2} \) |
| 61 | \( 1 + (-1.22e5 + 7.04e4i)T + (1.57e12 - 2.72e12i)T^{2} \) |
| 67 | \( 1 + (-2.07e4 - 3.58e4i)T + (-3.03e12 + 5.24e12i)T^{2} \) |
| 71 | \( 1 + (2.70e6 + 4.69e6i)T + (-4.54e12 + 7.87e12i)T^{2} \) |
| 73 | \( 1 - 4.23e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + (3.07e6 - 1.77e6i)T + (9.60e12 - 1.66e13i)T^{2} \) |
| 83 | \( 1 + (4.14e6 - 7.17e6i)T + (-1.35e13 - 2.35e13i)T^{2} \) |
| 89 | \( 1 + (-1.91e6 - 1.10e6i)T + (2.21e13 + 3.83e13i)T^{2} \) |
| 97 | \( 1 - 1.27e7iT - 8.07e13T^{2} \) |
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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
−14.71901981601959810167500323739, −12.28946275236587546518571937428, −11.68452745122267300443093673183, −10.51026276561404114924840399620, −9.265737979390791116542790622843, −8.260485712253652345330469054466, −6.91061436746444251533152746693, −3.86136885582630282991955269369, −2.16017202802408584746830765887, −0.15785619165837812914519480894,
1.36442264203409582149018693599, 4.61999681337746199204346269235, 6.86056324641798388474870404976, 7.82193473047544052603646438340, 8.647663965879014003377710660245, 10.17910752262218705187495959797, 11.28626505028758831415277034578, 13.03266390569279085576110369029, 14.72516730642107356227392769383, 15.73511854783408420600846940926
