| 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 | \( 1 + (11.8 - 3.83i)T + (78.4 - 57.0i)T^{2} \) |
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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
−11.70550218712767180619202858440, −10.92824999303268372666807810845, −10.63547309359861802441393172601, −8.854637113742703598302795984783, −8.402802707290839807360758827454, −6.93004876080285164876168157757, −5.76083347481761782333943689933, −4.83299556015408331314571594543, −3.72837704072875898612648123272, −2.40701306617429839314108073944,
0.05685392760362440833075760594, 2.90875204376015646479151389359, 4.24666054537160279790794521748, 5.14674633019978359845385212112, 6.37536453590352547877118804524, 7.20533346444828536394986513604, 8.147350799615781201617633099961, 8.949737488479646424963330441844, 10.60815625787194462041062754350, 11.13200221446037136422153435140
