| L(s) = 1 | + (1.20 + 1.44i)2-s + (6.02 + 5.05i)3-s + (0.774 − 4.39i)4-s + (−1.51 − 4.14i)5-s + 14.7i·6-s + (−23.8 + 8.67i)7-s + (20.3 − 11.7i)8-s + (6.04 + 34.3i)9-s + (4.15 − 7.19i)10-s + (−11.9 − 20.7i)11-s + (26.8 − 22.5i)12-s + (19.1 + 3.37i)13-s + (−41.3 − 23.8i)14-s + (11.8 − 32.6i)15-s + (7.89 + 2.87i)16-s + (−89.7 + 15.8i)17-s + ⋯ |
| L(s) = 1 | + (0.427 + 0.509i)2-s + (1.15 + 0.972i)3-s + (0.0968 − 0.549i)4-s + (−0.135 − 0.371i)5-s + 1.00i·6-s + (−1.28 + 0.468i)7-s + (0.897 − 0.517i)8-s + (0.224 + 1.27i)9-s + (0.131 − 0.227i)10-s + (−0.328 − 0.569i)11-s + (0.646 − 0.542i)12-s + (0.408 + 0.0719i)13-s + (−0.788 − 0.455i)14-s + (0.204 − 0.561i)15-s + (0.123 + 0.0448i)16-s + (−1.28 + 0.225i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.656 - 0.754i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.656 - 0.754i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.76543 + 0.804100i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.76543 + 0.804100i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 37 | \( 1 + (-28.3 - 223. i)T \) |
| good | 2 | \( 1 + (-1.20 - 1.44i)T + (-1.38 + 7.87i)T^{2} \) |
| 3 | \( 1 + (-6.02 - 5.05i)T + (4.68 + 26.5i)T^{2} \) |
| 5 | \( 1 + (1.51 + 4.14i)T + (-95.7 + 80.3i)T^{2} \) |
| 7 | \( 1 + (23.8 - 8.67i)T + (262. - 220. i)T^{2} \) |
| 11 | \( 1 + (11.9 + 20.7i)T + (-665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 + (-19.1 - 3.37i)T + (2.06e3 + 751. i)T^{2} \) |
| 17 | \( 1 + (89.7 - 15.8i)T + (4.61e3 - 1.68e3i)T^{2} \) |
| 19 | \( 1 + (-53.3 + 63.5i)T + (-1.19e3 - 6.75e3i)T^{2} \) |
| 23 | \( 1 + (-93.7 - 54.1i)T + (6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 + (143. - 82.6i)T + (1.21e4 - 2.11e4i)T^{2} \) |
| 31 | \( 1 - 269. iT - 2.97e4T^{2} \) |
| 41 | \( 1 + (77.6 - 440. i)T + (-6.47e4 - 2.35e4i)T^{2} \) |
| 43 | \( 1 + 352. iT - 7.95e4T^{2} \) |
| 47 | \( 1 + (-72.7 + 126. i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + (333. + 121. i)T + (1.14e5 + 9.56e4i)T^{2} \) |
| 59 | \( 1 + (79.6 - 218. i)T + (-1.57e5 - 1.32e5i)T^{2} \) |
| 61 | \( 1 + (362. + 63.8i)T + (2.13e5 + 7.76e4i)T^{2} \) |
| 67 | \( 1 + (-788. + 286. i)T + (2.30e5 - 1.93e5i)T^{2} \) |
| 71 | \( 1 + (124. + 104. i)T + (6.21e4 + 3.52e5i)T^{2} \) |
| 73 | \( 1 - 1.18e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + (112. + 307. i)T + (-3.77e5 + 3.16e5i)T^{2} \) |
| 83 | \( 1 + (177. + 1.00e3i)T + (-5.37e5 + 1.95e5i)T^{2} \) |
| 89 | \( 1 + (253. - 697. i)T + (-5.40e5 - 4.53e5i)T^{2} \) |
| 97 | \( 1 + (154. + 89.0i)T + (4.56e5 + 7.90e5i)T^{2} \) |
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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
−15.71337327463462350937061949262, −15.18599536056724374710809203972, −13.83918010645114408631837527736, −13.04825502521510767419722377427, −10.83038154250462791003884597266, −9.544184539905654063316614830372, −8.703405969146878623002745173903, −6.64192307556873100213825763086, −4.93089787541461505703530695706, −3.22134723036410837326800265072,
2.47756545399451007354025729448, 3.70642805122189636792455329801, 6.86005166272793124073454431149, 7.75677083672090673039406951230, 9.300538994736342793555990374004, 11.03362141750138323448240507474, 12.72381474323749630883226383002, 13.09624394283455326701190616320, 14.08971608524327094190381796194, 15.53251847472219053893515313333
