| L(s) = 1 | + (0.502 − 0.0885i)2-s + (0.199 − 1.13i)3-s + (−1.63 + 0.595i)4-s + (−0.986 + 1.17i)5-s − 0.587i·6-s + (0.422 + 0.354i)7-s + (−1.65 + 0.954i)8-s + (1.57 + 0.572i)9-s + (−0.391 + 0.678i)10-s + (−2.20 − 3.82i)11-s + (0.347 + 1.97i)12-s + (−0.881 − 2.42i)13-s + (0.243 + 0.140i)14-s + (1.13 + 1.35i)15-s + (1.91 − 1.61i)16-s + (−1.25 + 3.43i)17-s + ⋯ |
| L(s) = 1 | + (0.355 − 0.0626i)2-s + (0.115 − 0.654i)3-s + (−0.817 + 0.297i)4-s + (−0.441 + 0.525i)5-s − 0.239i·6-s + (0.159 + 0.134i)7-s + (−0.584 + 0.337i)8-s + (0.524 + 0.190i)9-s + (−0.123 + 0.214i)10-s + (−0.665 − 1.15i)11-s + (0.100 + 0.569i)12-s + (−0.244 − 0.671i)13-s + (0.0651 + 0.0376i)14-s + (0.293 + 0.349i)15-s + (0.479 − 0.402i)16-s + (−0.303 + 0.834i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.972 + 0.232i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.972 + 0.232i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.760151 - 0.0895545i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.760151 - 0.0895545i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 37 | \( 1 + (4.52 - 4.06i)T \) |
| good | 2 | \( 1 + (-0.502 + 0.0885i)T + (1.87 - 0.684i)T^{2} \) |
| 3 | \( 1 + (-0.199 + 1.13i)T + (-2.81 - 1.02i)T^{2} \) |
| 5 | \( 1 + (0.986 - 1.17i)T + (-0.868 - 4.92i)T^{2} \) |
| 7 | \( 1 + (-0.422 - 0.354i)T + (1.21 + 6.89i)T^{2} \) |
| 11 | \( 1 + (2.20 + 3.82i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (0.881 + 2.42i)T + (-9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (1.25 - 3.43i)T + (-13.0 - 10.9i)T^{2} \) |
| 19 | \( 1 + (-7.27 - 1.28i)T + (17.8 + 6.49i)T^{2} \) |
| 23 | \( 1 + (3.41 + 1.97i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (5.82 - 3.36i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 - 2.56iT - 31T^{2} \) |
| 41 | \( 1 + (0.0161 - 0.00589i)T + (31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + 6.06iT - 43T^{2} \) |
| 47 | \( 1 + (2.31 - 4.01i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-9.18 + 7.70i)T + (9.20 - 52.1i)T^{2} \) |
| 59 | \( 1 + (-5.30 - 6.32i)T + (-10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (-3.39 - 9.33i)T + (-46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (11.6 + 9.73i)T + (11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (-0.229 + 1.30i)T + (-66.7 - 24.2i)T^{2} \) |
| 73 | \( 1 + 0.316T + 73T^{2} \) |
| 79 | \( 1 + (-2.05 + 2.44i)T + (-13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (7.17 + 2.61i)T + (63.5 + 53.3i)T^{2} \) |
| 89 | \( 1 + (-6.14 - 7.32i)T + (-15.4 + 87.6i)T^{2} \) |
| 97 | \( 1 + (-1.74 - 1.00i)T + (48.5 + 84.0i)T^{2} \) |
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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
−16.34602987761318620412615324254, −15.02785469385061043786778501383, −13.79056134526119714614896657562, −13.01970467089825937009434617386, −11.84530230661970265142518809214, −10.32962189299348175251013855331, −8.489865306907854070038509278956, −7.46306529988366611033552385764, −5.45337965994512139835849049778, −3.41365332415871423940694010008,
4.14979624831693860514966124349, 5.09304116028829617481927488170, 7.45994281634412123223063313067, 9.259524001570439782348004411582, 9.955179469288874424764751791518, 11.84495689970496015375237596124, 13.02608723580589825027751073339, 14.19468363501308151702217908796, 15.38514245132069705633720664276, 16.11597085103720905571827853694
