| L(s) = 1 | + (3.09 + 3.69i)2-s + (1.24 + 1.04i)3-s + (−2.64 + 15.0i)4-s + (−4.06 − 11.1i)5-s + 7.80i·6-s + (6.95 − 2.53i)7-s + (−30.3 + 17.5i)8-s + (−4.23 − 24.0i)9-s + (28.6 − 49.6i)10-s + (25.0 + 43.4i)11-s + (−18.9 + 15.8i)12-s + (−55.0 − 9.71i)13-s + (30.9 + 17.8i)14-s + (6.58 − 18.0i)15-s + (−43.9 − 15.9i)16-s + (25.6 − 4.51i)17-s + ⋯ |
| L(s) = 1 | + (1.09 + 1.30i)2-s + (0.238 + 0.200i)3-s + (−0.331 + 1.87i)4-s + (−0.363 − 0.999i)5-s + 0.531i·6-s + (0.375 − 0.136i)7-s + (−1.33 + 0.773i)8-s + (−0.156 − 0.889i)9-s + (0.906 − 1.57i)10-s + (0.687 + 1.19i)11-s + (−0.455 + 0.381i)12-s + (−1.17 − 0.207i)13-s + (0.590 + 0.340i)14-s + (0.113 − 0.311i)15-s + (−0.686 − 0.249i)16-s + (0.365 − 0.0644i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0403 - 0.999i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.0403 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.51664 + 1.45669i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.51664 + 1.45669i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 37 | \( 1 + (-161. - 156. i)T \) |
| good | 2 | \( 1 + (-3.09 - 3.69i)T + (-1.38 + 7.87i)T^{2} \) |
| 3 | \( 1 + (-1.24 - 1.04i)T + (4.68 + 26.5i)T^{2} \) |
| 5 | \( 1 + (4.06 + 11.1i)T + (-95.7 + 80.3i)T^{2} \) |
| 7 | \( 1 + (-6.95 + 2.53i)T + (262. - 220. i)T^{2} \) |
| 11 | \( 1 + (-25.0 - 43.4i)T + (-665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 + (55.0 + 9.71i)T + (2.06e3 + 751. i)T^{2} \) |
| 17 | \( 1 + (-25.6 + 4.51i)T + (4.61e3 - 1.68e3i)T^{2} \) |
| 19 | \( 1 + (-14.9 + 17.7i)T + (-1.19e3 - 6.75e3i)T^{2} \) |
| 23 | \( 1 + (149. + 86.2i)T + (6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 + (223. - 129. i)T + (1.21e4 - 2.11e4i)T^{2} \) |
| 31 | \( 1 - 180. iT - 2.97e4T^{2} \) |
| 41 | \( 1 + (-66.4 + 377. i)T + (-6.47e4 - 2.35e4i)T^{2} \) |
| 43 | \( 1 - 254. iT - 7.95e4T^{2} \) |
| 47 | \( 1 + (3.64 - 6.30i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + (-263. - 96.0i)T + (1.14e5 + 9.56e4i)T^{2} \) |
| 59 | \( 1 + (-264. + 727. i)T + (-1.57e5 - 1.32e5i)T^{2} \) |
| 61 | \( 1 + (-385. - 67.9i)T + (2.13e5 + 7.76e4i)T^{2} \) |
| 67 | \( 1 + (-599. + 218. i)T + (2.30e5 - 1.93e5i)T^{2} \) |
| 71 | \( 1 + (211. + 177. i)T + (6.21e4 + 3.52e5i)T^{2} \) |
| 73 | \( 1 - 711.T + 3.89e5T^{2} \) |
| 79 | \( 1 + (-222. - 610. i)T + (-3.77e5 + 3.16e5i)T^{2} \) |
| 83 | \( 1 + (92.7 + 525. i)T + (-5.37e5 + 1.95e5i)T^{2} \) |
| 89 | \( 1 + (301. - 829. i)T + (-5.40e5 - 4.53e5i)T^{2} \) |
| 97 | \( 1 + (-309. - 178. i)T + (4.56e5 + 7.90e5i)T^{2} \) |
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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
−15.95677635272837442365364710717, −14.82412555718043522803903537350, −14.34017440151068509221013608692, −12.58942448084247505950802654377, −12.17581139857070478444625456290, −9.520427835577477268546567982001, −8.109032245279719687332095011523, −6.85853055596279450375980927787, −5.10508795279640442187220352110, −4.06088477567343091260194570776,
2.31925386969434378862787628741, 3.80843654105025230387801254491, 5.65306066606119608655249875615, 7.72909010794453009143074320560, 9.884599712882122101972563730731, 11.22731083480443122362707123679, 11.70135568711243519846997366880, 13.27576963114491812103945044402, 14.24144713558536653740921252037, 14.83903767857552348081226942894
