| L(s) = 1 | + (0.266 + 1.50i)3-s + (−0.247 − 0.294i)5-s + (2.50 − 2.09i)7-s + (0.613 − 0.223i)9-s + (1.29 − 2.23i)11-s + (−0.466 + 1.28i)13-s + (0.378 − 0.451i)15-s + (−1.13 − 3.13i)17-s + (5.89 − 1.03i)19-s + (3.83 + 3.21i)21-s + (−6.53 + 3.77i)23-s + (0.842 − 4.77i)25-s + (2.79 + 4.84i)27-s + (2.78 + 1.60i)29-s − 2.53i·31-s + ⋯ |
| L(s) = 1 | + (0.153 + 0.871i)3-s + (−0.110 − 0.131i)5-s + (0.945 − 0.793i)7-s + (0.204 − 0.0744i)9-s + (0.389 − 0.674i)11-s + (−0.129 + 0.355i)13-s + (0.0978 − 0.116i)15-s + (−0.276 − 0.759i)17-s + (1.35 − 0.238i)19-s + (0.835 + 0.701i)21-s + (−1.36 + 0.786i)23-s + (0.168 − 0.955i)25-s + (0.538 + 0.932i)27-s + (0.516 + 0.298i)29-s − 0.455i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.983 - 0.181i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 592 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.983 - 0.181i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.75896 + 0.161287i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.75896 + 0.161287i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 37 | \( 1 + (-0.543 - 6.05i)T \) |
| good | 3 | \( 1 + (-0.266 - 1.50i)T + (-2.81 + 1.02i)T^{2} \) |
| 5 | \( 1 + (0.247 + 0.294i)T + (-0.868 + 4.92i)T^{2} \) |
| 7 | \( 1 + (-2.50 + 2.09i)T + (1.21 - 6.89i)T^{2} \) |
| 11 | \( 1 + (-1.29 + 2.23i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (0.466 - 1.28i)T + (-9.95 - 8.35i)T^{2} \) |
| 17 | \( 1 + (1.13 + 3.13i)T + (-13.0 + 10.9i)T^{2} \) |
| 19 | \( 1 + (-5.89 + 1.03i)T + (17.8 - 6.49i)T^{2} \) |
| 23 | \( 1 + (6.53 - 3.77i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-2.78 - 1.60i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 2.53iT - 31T^{2} \) |
| 41 | \( 1 + (-7.77 - 2.82i)T + (31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 - 4.33iT - 43T^{2} \) |
| 47 | \( 1 + (-2.61 - 4.52i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (6.64 + 5.57i)T + (9.20 + 52.1i)T^{2} \) |
| 59 | \( 1 + (8.33 - 9.93i)T + (-10.2 - 58.1i)T^{2} \) |
| 61 | \( 1 + (2.39 - 6.58i)T + (-46.7 - 39.2i)T^{2} \) |
| 67 | \( 1 + (-8.60 + 7.22i)T + (11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (2.60 + 14.7i)T + (-66.7 + 24.2i)T^{2} \) |
| 73 | \( 1 + 15.0T + 73T^{2} \) |
| 79 | \( 1 + (0.940 + 1.12i)T + (-13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (0.0104 - 0.00380i)T + (63.5 - 53.3i)T^{2} \) |
| 89 | \( 1 + (0.612 - 0.730i)T + (-15.4 - 87.6i)T^{2} \) |
| 97 | \( 1 + (-12.8 + 7.40i)T + (48.5 - 84.0i)T^{2} \) |
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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
−10.68161997414417177428825049207, −9.781841858434573705218382813704, −9.170471923208156853435503451273, −8.040031150018102536418695091484, −7.33191706523719452582941277120, −6.09347074462417269057782044425, −4.76498065639108157895875895659, −4.28508213820424301953234445666, −3.11910983607662239642099767756, −1.23573998825152571009206056160,
1.49179428884758878541173286231, 2.41933277031499739236793763235, 4.03969817352564431575849905582, 5.18711959434593714806595615329, 6.18719813229161612335321769277, 7.29016149733440609916422419631, 7.87589934165886412252196119148, 8.716164661523374078985253982902, 9.766296287405475293252380573736, 10.72174990021229524082853810982
