| L(s) = 1 | + (0.909 − 0.415i)2-s + (−0.537 − 1.83i)3-s + (0.654 − 0.755i)4-s + (1.72 − 1.42i)5-s + (−1.24 − 1.44i)6-s + (−2.93 + 0.422i)7-s + (0.281 − 0.959i)8-s + (−0.537 + 0.345i)9-s + (0.979 − 2.01i)10-s + (−1.22 + 2.68i)11-s + (−1.73 − 0.792i)12-s + (1.87 + 0.269i)13-s + (−2.49 + 1.60i)14-s + (−3.52 − 2.39i)15-s + (−0.142 − 0.989i)16-s + (0.719 − 0.623i)17-s + ⋯ |
| L(s) = 1 | + (0.643 − 0.293i)2-s + (−0.310 − 1.05i)3-s + (0.327 − 0.377i)4-s + (0.771 − 0.635i)5-s + (−0.510 − 0.588i)6-s + (−1.10 + 0.159i)7-s + (0.0996 − 0.339i)8-s + (−0.179 + 0.115i)9-s + (0.309 − 0.635i)10-s + (−0.370 + 0.810i)11-s + (−0.500 − 0.228i)12-s + (0.520 + 0.0748i)13-s + (−0.666 + 0.428i)14-s + (−0.911 − 0.618i)15-s + (−0.0355 − 0.247i)16-s + (0.174 − 0.151i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.188 + 0.982i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.188 + 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.03564 - 1.25343i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.03564 - 1.25343i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.909 + 0.415i)T \) |
| 5 | \( 1 + (-1.72 + 1.42i)T \) |
| 23 | \( 1 + (-4.04 - 2.57i)T \) |
| good | 3 | \( 1 + (0.537 + 1.83i)T + (-2.52 + 1.62i)T^{2} \) |
| 7 | \( 1 + (2.93 - 0.422i)T + (6.71 - 1.97i)T^{2} \) |
| 11 | \( 1 + (1.22 - 2.68i)T + (-7.20 - 8.31i)T^{2} \) |
| 13 | \( 1 + (-1.87 - 0.269i)T + (12.4 + 3.66i)T^{2} \) |
| 17 | \( 1 + (-0.719 + 0.623i)T + (2.41 - 16.8i)T^{2} \) |
| 19 | \( 1 + (-0.167 + 0.193i)T + (-2.70 - 18.8i)T^{2} \) |
| 29 | \( 1 + (0.0268 + 0.0309i)T + (-4.12 + 28.7i)T^{2} \) |
| 31 | \( 1 + (-5.13 - 1.50i)T + (26.0 + 16.7i)T^{2} \) |
| 37 | \( 1 + (2.00 + 3.12i)T + (-15.3 + 33.6i)T^{2} \) |
| 41 | \( 1 + (-7.88 - 5.06i)T + (17.0 + 37.2i)T^{2} \) |
| 43 | \( 1 + (-3.10 - 10.5i)T + (-36.1 + 23.2i)T^{2} \) |
| 47 | \( 1 - 8.07iT - 47T^{2} \) |
| 53 | \( 1 + (12.5 - 1.79i)T + (50.8 - 14.9i)T^{2} \) |
| 59 | \( 1 + (-0.794 + 5.52i)T + (-56.6 - 16.6i)T^{2} \) |
| 61 | \( 1 + (-8.08 - 2.37i)T + (51.3 + 32.9i)T^{2} \) |
| 67 | \( 1 + (5.22 - 2.38i)T + (43.8 - 50.6i)T^{2} \) |
| 71 | \( 1 + (5.95 + 13.0i)T + (-46.4 + 53.6i)T^{2} \) |
| 73 | \( 1 + (0.785 + 0.680i)T + (10.3 + 72.2i)T^{2} \) |
| 79 | \( 1 + (0.123 - 0.862i)T + (-75.7 - 22.2i)T^{2} \) |
| 83 | \( 1 + (6.11 + 9.51i)T + (-34.4 + 75.4i)T^{2} \) |
| 89 | \( 1 + (15.5 - 4.57i)T + (74.8 - 48.1i)T^{2} \) |
| 97 | \( 1 + (1.81 - 2.82i)T + (-40.2 - 88.2i)T^{2} \) |
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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
−12.45150697927131105974192633718, −11.26149628141366117380874010291, −9.942516943640751173238921283565, −9.275711802093569535089888175518, −7.67286372923123403363177714913, −6.53438461201858930886815507304, −5.92649098855171660614383311601, −4.62017341874472210071878164358, −2.83291435471286779796623215537, −1.34625849353527549663225715389,
2.89764920650097295690137922768, 3.88379458786060660051624586617, 5.32593195695556336874710061579, 6.12313433265703571029292495884, 7.09447614302129838401521041205, 8.722313926264157611366890575350, 9.861444161531175797800576788036, 10.51851371211801080541848547649, 11.28246383874313366942309742917, 12.74431363573678261066151503085
