| 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 | \( 1 + (-738. + 426. i)T + (4.56e5 - 7.90e5i)T^{2} \) |
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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
−15.71609570564528371667987669284, −13.91183964749251745854538414243, −12.89750958396243284993127709824, −12.42958424195509763827106617728, −10.87769381528596005023324115290, −9.741810858101865686865873634271, −8.050760912082039785979029580806, −5.28907141281185423836403523856, −4.56823947985406253012152580948, −1.97981507881818476487619555552,
3.37297681560347022816475181043, 5.97562113549336698186138550479, 6.29130967474514084446571310197, 7.88532921979714434058272713718, 10.15406117929852572142409382138, 11.24996429071312390997129728685, 13.05176415291696915235335144604, 14.09795553930660031240990495247, 14.74661668474875328268808987816, 15.88537484300134472484636036549
