| L(s) = 1 | + (−0.142 + 0.989i)2-s + (−0.844 − 1.85i)3-s + (−0.959 − 0.281i)4-s + (0.654 + 0.755i)5-s + (1.95 − 0.573i)6-s + (1.31 + 0.847i)7-s + (0.415 − 0.909i)8-s + (−0.744 + 0.859i)9-s + (−0.841 + 0.540i)10-s + (−0.774 − 5.38i)11-s + (0.289 + 2.01i)12-s + (2.91 − 1.87i)13-s + (−1.02 + 1.18i)14-s + (0.844 − 1.85i)15-s + (0.841 + 0.540i)16-s + (7.30 − 2.14i)17-s + ⋯ |
| L(s) = 1 | + (−0.100 + 0.699i)2-s + (−0.487 − 1.06i)3-s + (−0.479 − 0.140i)4-s + (0.292 + 0.337i)5-s + (0.796 − 0.233i)6-s + (0.498 + 0.320i)7-s + (0.146 − 0.321i)8-s + (−0.248 + 0.286i)9-s + (−0.266 + 0.170i)10-s + (−0.233 − 1.62i)11-s + (0.0835 + 0.581i)12-s + (0.808 − 0.519i)13-s + (−0.274 + 0.316i)14-s + (0.218 − 0.477i)15-s + (0.210 + 0.135i)16-s + (1.77 − 0.520i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.886 + 0.461i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.886 + 0.461i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.03203 - 0.252688i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.03203 - 0.252688i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.142 - 0.989i)T \) |
| 5 | \( 1 + (-0.654 - 0.755i)T \) |
| 23 | \( 1 + (-2.22 - 4.24i)T \) |
| good | 3 | \( 1 + (0.844 + 1.85i)T + (-1.96 + 2.26i)T^{2} \) |
| 7 | \( 1 + (-1.31 - 0.847i)T + (2.90 + 6.36i)T^{2} \) |
| 11 | \( 1 + (0.774 + 5.38i)T + (-10.5 + 3.09i)T^{2} \) |
| 13 | \( 1 + (-2.91 + 1.87i)T + (5.40 - 11.8i)T^{2} \) |
| 17 | \( 1 + (-7.30 + 2.14i)T + (14.3 - 9.19i)T^{2} \) |
| 19 | \( 1 + (4.23 + 1.24i)T + (15.9 + 10.2i)T^{2} \) |
| 29 | \( 1 + (0.703 - 0.206i)T + (24.3 - 15.6i)T^{2} \) |
| 31 | \( 1 + (1.02 - 2.25i)T + (-20.3 - 23.4i)T^{2} \) |
| 37 | \( 1 + (7.09 - 8.18i)T + (-5.26 - 36.6i)T^{2} \) |
| 41 | \( 1 + (-1.76 - 2.04i)T + (-5.83 + 40.5i)T^{2} \) |
| 43 | \( 1 + (0.873 + 1.91i)T + (-28.1 + 32.4i)T^{2} \) |
| 47 | \( 1 - 7.35T + 47T^{2} \) |
| 53 | \( 1 + (-4.95 - 3.18i)T + (22.0 + 48.2i)T^{2} \) |
| 59 | \( 1 + (-2.19 + 1.41i)T + (24.5 - 53.6i)T^{2} \) |
| 61 | \( 1 + (5.24 - 11.4i)T + (-39.9 - 46.1i)T^{2} \) |
| 67 | \( 1 + (-0.481 + 3.34i)T + (-64.2 - 18.8i)T^{2} \) |
| 71 | \( 1 + (-0.270 + 1.88i)T + (-68.1 - 20.0i)T^{2} \) |
| 73 | \( 1 + (6.55 + 1.92i)T + (61.4 + 39.4i)T^{2} \) |
| 79 | \( 1 + (1.33 - 0.859i)T + (32.8 - 71.8i)T^{2} \) |
| 83 | \( 1 + (9.05 - 10.4i)T + (-11.8 - 82.1i)T^{2} \) |
| 89 | \( 1 + (6.11 + 13.4i)T + (-58.2 + 67.2i)T^{2} \) |
| 97 | \( 1 + (-10.1 - 11.6i)T + (-13.8 + 96.0i)T^{2} \) |
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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
−12.17072812830901357894360663531, −11.28336974280719463395429877642, −10.29997019519400650493151095865, −8.824807300150102157696159018017, −8.017556174683981184648799947540, −7.05211990098231598296097464736, −5.94672221405673598358222570145, −5.46001652161261595725195279409, −3.29680976047071709639245378431, −1.13589530683588642467680767420,
1.80869299874966810139307959799, 3.91511007053132066120440848213, 4.64761063405101600389980643953, 5.70999376075729274333836026547, 7.42501372978936051115243786149, 8.679111584385530160691210882259, 9.752104695009229076477547220793, 10.37174793434961772771314161953, 11.03500199988004880365175320529, 12.25955040579279118129001528879
