| L(s) = 1 | + (0.587 + 0.809i)2-s + (−0.855 − 0.277i)3-s + (−0.309 + 0.951i)4-s + (−2.23 − 0.00941i)5-s + (−0.277 − 0.855i)6-s − i·7-s + (−0.951 + 0.309i)8-s + (−1.77 − 1.28i)9-s + (−1.30 − 1.81i)10-s + (−3.98 + 2.89i)11-s + (0.528 − 0.727i)12-s + (2.51 − 3.46i)13-s + (0.809 − 0.587i)14-s + (1.91 + 0.629i)15-s + (−0.809 − 0.587i)16-s + (−7.39 + 2.40i)17-s + ⋯ |
| L(s) = 1 | + (0.415 + 0.572i)2-s + (−0.493 − 0.160i)3-s + (−0.154 + 0.475i)4-s + (−0.999 − 0.00421i)5-s + (−0.113 − 0.349i)6-s − 0.377i·7-s + (−0.336 + 0.109i)8-s + (−0.590 − 0.429i)9-s + (−0.413 − 0.573i)10-s + (−1.20 + 0.872i)11-s + (0.152 − 0.210i)12-s + (0.698 − 0.961i)13-s + (0.216 − 0.157i)14-s + (0.493 + 0.162i)15-s + (−0.202 − 0.146i)16-s + (−1.79 + 0.582i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.734 + 0.678i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.734 + 0.678i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0344542 - 0.0881038i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0344542 - 0.0881038i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.587 - 0.809i)T \) |
| 5 | \( 1 + (2.23 + 0.00941i)T \) |
| 7 | \( 1 + iT \) |
| good | 3 | \( 1 + (0.855 + 0.277i)T + (2.42 + 1.76i)T^{2} \) |
| 11 | \( 1 + (3.98 - 2.89i)T + (3.39 - 10.4i)T^{2} \) |
| 13 | \( 1 + (-2.51 + 3.46i)T + (-4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (7.39 - 2.40i)T + (13.7 - 9.99i)T^{2} \) |
| 19 | \( 1 + (1.65 + 5.07i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + (-1.17 - 1.62i)T + (-7.10 + 21.8i)T^{2} \) |
| 29 | \( 1 + (-0.306 + 0.943i)T + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (1.36 + 4.20i)T + (-25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (5.82 - 8.02i)T + (-11.4 - 35.1i)T^{2} \) |
| 41 | \( 1 + (4.71 + 3.42i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 - 6.77iT - 43T^{2} \) |
| 47 | \( 1 + (-10.7 - 3.49i)T + (38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (2.24 + 0.730i)T + (42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (-3.42 - 2.48i)T + (18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (2.23 - 1.62i)T + (18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + (5.81 - 1.88i)T + (54.2 - 39.3i)T^{2} \) |
| 71 | \( 1 + (-2.95 + 9.10i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (3.37 + 4.64i)T + (-22.5 + 69.4i)T^{2} \) |
| 79 | \( 1 + (0.680 - 2.09i)T + (-63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (-2.47 + 0.804i)T + (67.1 - 48.7i)T^{2} \) |
| 89 | \( 1 + (-6.01 + 4.36i)T + (27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (11.8 + 3.83i)T + (78.4 + 57.0i)T^{2} \) |
| 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
−11.13200221446037136422153435140, −10.60815625787194462041062754350, −8.949737488479646424963330441844, −8.147350799615781201617633099961, −7.20533346444828536394986513604, −6.37536453590352547877118804524, −5.14674633019978359845385212112, −4.24666054537160279790794521748, −2.90875204376015646479151389359, −0.05685392760362440833075760594,
2.40701306617429839314108073944, 3.72837704072875898612648123272, 4.83299556015408331314571594543, 5.76083347481761782333943689933, 6.93004876080285164876168157757, 8.402802707290839807360758827454, 8.854637113742703598302795984783, 10.63547309359861802441393172601, 10.92824999303268372666807810845, 11.70550218712767180619202858440
