| L(s) = 1 | + (2.37 + 2.82i)2-s + (−1.46 − 1.22i)3-s + (−0.975 + 5.53i)4-s + (6.54 + 17.9i)5-s − 7.04i·6-s + (−0.795 + 0.289i)7-s + (7.61 − 4.39i)8-s + (−4.05 − 23.0i)9-s + (−35.3 + 61.2i)10-s + (−32.8 − 56.9i)11-s + (8.21 − 6.89i)12-s + (22.6 + 3.98i)13-s + (−2.70 − 1.56i)14-s + (12.5 − 34.3i)15-s + (72.7 + 26.4i)16-s + (42.6 − 7.51i)17-s + ⋯ |
| L(s) = 1 | + (0.838 + 0.999i)2-s + (−0.281 − 0.236i)3-s + (−0.121 + 0.691i)4-s + (0.585 + 1.60i)5-s − 0.479i·6-s + (−0.0429 + 0.0156i)7-s + (0.336 − 0.194i)8-s + (−0.150 − 0.851i)9-s + (−1.11 + 1.93i)10-s + (−0.901 − 1.56i)11-s + (0.197 − 0.165i)12-s + (0.482 + 0.0850i)13-s + (−0.0516 − 0.0298i)14-s + (0.215 − 0.591i)15-s + (1.13 + 0.413i)16-s + (0.608 − 0.107i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.241 - 0.970i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.241 - 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.44319 + 1.12845i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.44319 + 1.12845i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 37 | \( 1 + (-126. - 185. i)T \) |
| good | 2 | \( 1 + (-2.37 - 2.82i)T + (-1.38 + 7.87i)T^{2} \) |
| 3 | \( 1 + (1.46 + 1.22i)T + (4.68 + 26.5i)T^{2} \) |
| 5 | \( 1 + (-6.54 - 17.9i)T + (-95.7 + 80.3i)T^{2} \) |
| 7 | \( 1 + (0.795 - 0.289i)T + (262. - 220. i)T^{2} \) |
| 11 | \( 1 + (32.8 + 56.9i)T + (-665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 + (-22.6 - 3.98i)T + (2.06e3 + 751. i)T^{2} \) |
| 17 | \( 1 + (-42.6 + 7.51i)T + (4.61e3 - 1.68e3i)T^{2} \) |
| 19 | \( 1 + (65.7 - 78.3i)T + (-1.19e3 - 6.75e3i)T^{2} \) |
| 23 | \( 1 + (58.0 + 33.5i)T + (6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 + (-4.90 + 2.83i)T + (1.21e4 - 2.11e4i)T^{2} \) |
| 31 | \( 1 + 195. iT - 2.97e4T^{2} \) |
| 41 | \( 1 + (-4.12 + 23.3i)T + (-6.47e4 - 2.35e4i)T^{2} \) |
| 43 | \( 1 - 166. iT - 7.95e4T^{2} \) |
| 47 | \( 1 + (-49.3 + 85.4i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + (643. + 234. i)T + (1.14e5 + 9.56e4i)T^{2} \) |
| 59 | \( 1 + (147. - 404. i)T + (-1.57e5 - 1.32e5i)T^{2} \) |
| 61 | \( 1 + (61.8 + 10.9i)T + (2.13e5 + 7.76e4i)T^{2} \) |
| 67 | \( 1 + (-335. + 121. i)T + (2.30e5 - 1.93e5i)T^{2} \) |
| 71 | \( 1 + (-743. - 623. i)T + (6.21e4 + 3.52e5i)T^{2} \) |
| 73 | \( 1 - 112.T + 3.89e5T^{2} \) |
| 79 | \( 1 + (252. + 693. i)T + (-3.77e5 + 3.16e5i)T^{2} \) |
| 83 | \( 1 + (122. + 696. i)T + (-5.37e5 + 1.95e5i)T^{2} \) |
| 89 | \( 1 + (-460. + 1.26e3i)T + (-5.40e5 - 4.53e5i)T^{2} \) |
| 97 | \( 1 + (-738. - 426. i)T + (4.56e5 + 7.90e5i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−15.88537484300134472484636036549, −14.74661668474875328268808987816, −14.09795553930660031240990495247, −13.05176415291696915235335144604, −11.24996429071312390997129728685, −10.15406117929852572142409382138, −7.88532921979714434058272713718, −6.29130967474514084446571310197, −5.97562113549336698186138550479, −3.37297681560347022816475181043,
1.97981507881818476487619555552, 4.56823947985406253012152580948, 5.28907141281185423836403523856, 8.050760912082039785979029580806, 9.741810858101865686865873634271, 10.87769381528596005023324115290, 12.42958424195509763827106617728, 12.89750958396243284993127709824, 13.91183964749251745854538414243, 15.71609570564528371667987669284
