| L(s) = 1 | + (−1.72 − 4.74i)2-s + (−3.76 − 1.37i)3-s + (−13.4 + 11.2i)4-s + (2.75 − 0.485i)5-s + 20.2i·6-s + (−0.954 − 5.41i)7-s + (41.5 + 23.9i)8-s + (−8.36 − 7.01i)9-s + (−7.06 − 12.2i)10-s + (−22.6 + 39.2i)11-s + (65.9 − 23.9i)12-s + (−56.4 − 67.3i)13-s + (−24.0 + 13.8i)14-s + (−11.0 − 1.94i)15-s + (17.7 − 100. i)16-s + (8.22 − 9.80i)17-s + ⋯ |
| L(s) = 1 | + (−0.610 − 1.67i)2-s + (−0.725 − 0.263i)3-s + (−1.67 + 1.40i)4-s + (0.246 − 0.0434i)5-s + 1.37i·6-s + (−0.0515 − 0.292i)7-s + (1.83 + 1.05i)8-s + (−0.309 − 0.259i)9-s + (−0.223 − 0.386i)10-s + (−0.621 + 1.07i)11-s + (1.58 − 0.577i)12-s + (−1.20 − 1.43i)13-s + (−0.459 + 0.265i)14-s + (−0.190 − 0.0335i)15-s + (0.277 − 1.57i)16-s + (0.117 − 0.139i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.594 - 0.804i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.594 - 0.804i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.172244 + 0.341373i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.172244 + 0.341373i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 37 | \( 1 + (-88.2 + 207. i)T \) |
| good | 2 | \( 1 + (1.72 + 4.74i)T + (-6.12 + 5.14i)T^{2} \) |
| 3 | \( 1 + (3.76 + 1.37i)T + (20.6 + 17.3i)T^{2} \) |
| 5 | \( 1 + (-2.75 + 0.485i)T + (117. - 42.7i)T^{2} \) |
| 7 | \( 1 + (0.954 + 5.41i)T + (-322. + 117. i)T^{2} \) |
| 11 | \( 1 + (22.6 - 39.2i)T + (-665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 + (56.4 + 67.3i)T + (-381. + 2.16e3i)T^{2} \) |
| 17 | \( 1 + (-8.22 + 9.80i)T + (-853. - 4.83e3i)T^{2} \) |
| 19 | \( 1 + (-36.4 + 100. i)T + (-5.25e3 - 4.40e3i)T^{2} \) |
| 23 | \( 1 + (-156. + 90.6i)T + (6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 + (98.5 + 56.8i)T + (1.21e4 + 2.11e4i)T^{2} \) |
| 31 | \( 1 + 22.5iT - 2.97e4T^{2} \) |
| 41 | \( 1 + (68.6 - 57.5i)T + (1.19e4 - 6.78e4i)T^{2} \) |
| 43 | \( 1 - 248. iT - 7.95e4T^{2} \) |
| 47 | \( 1 + (74.2 + 128. i)T + (-5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 + (42.9 - 243. i)T + (-1.39e5 - 5.09e4i)T^{2} \) |
| 59 | \( 1 + (-176. - 31.1i)T + (1.92e5 + 7.02e4i)T^{2} \) |
| 61 | \( 1 + (380. + 453. i)T + (-3.94e4 + 2.23e5i)T^{2} \) |
| 67 | \( 1 + (33.3 + 189. i)T + (-2.82e5 + 1.02e5i)T^{2} \) |
| 71 | \( 1 + (-484. - 176. i)T + (2.74e5 + 2.30e5i)T^{2} \) |
| 73 | \( 1 + 421.T + 3.89e5T^{2} \) |
| 79 | \( 1 + (1.10e3 - 194. i)T + (4.63e5 - 1.68e5i)T^{2} \) |
| 83 | \( 1 + (-80.2 - 67.3i)T + (9.92e4 + 5.63e5i)T^{2} \) |
| 89 | \( 1 + (-1.33e3 - 234. i)T + (6.62e5 + 2.41e5i)T^{2} \) |
| 97 | \( 1 + (-443. + 256. i)T + (4.56e5 - 7.90e5i)T^{2} \) |
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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
−15.05899838549195702873387469759, −13.11843612112301339761914936375, −12.50142073018378689961168306333, −11.35049463334144420497011890817, −10.32435503532864028483946228301, −9.318729991996648677913033620439, −7.49688795760181908648483183047, −5.03351455886339043322315477194, −2.72589509225973397525777172491, −0.42767035751721293843031384167,
5.12218133431618920339981049190, 6.03144245309328299359913809535, 7.50101103697253953205905029896, 8.872284091064541582227878242102, 10.08945502722311442208013323209, 11.65254390247167638615240480205, 13.67141270096984711551404363713, 14.59379816656732551240793981898, 15.87859162377675826723962830048, 16.72343454381299739606887179578
