| L(s) = 1 | + (0.486 − 1.32i)2-s + (1.18 + 1.26i)3-s + (−1.52 − 1.29i)4-s + (−1.96 − 2.93i)5-s + (2.25 − 0.957i)6-s + (1.41 − 3.41i)7-s + (−2.45 + 1.40i)8-s + (−0.198 + 2.99i)9-s + (−4.85 + 1.17i)10-s + (1.35 + 0.269i)11-s + (−0.175 − 3.45i)12-s + (3.69 + 2.47i)13-s + (−3.84 − 3.53i)14-s + (1.39 − 5.96i)15-s + (0.666 + 3.94i)16-s + (−1.07 − 1.07i)17-s + ⋯ |
| L(s) = 1 | + (0.343 − 0.939i)2-s + (0.683 + 0.730i)3-s + (−0.763 − 0.645i)4-s + (−0.877 − 1.31i)5-s + (0.920 − 0.390i)6-s + (0.534 − 1.28i)7-s + (−0.868 + 0.495i)8-s + (−0.0662 + 0.997i)9-s + (−1.53 + 0.372i)10-s + (0.408 + 0.0812i)11-s + (−0.0505 − 0.998i)12-s + (1.02 + 0.685i)13-s + (−1.02 − 0.944i)14-s + (0.359 − 1.53i)15-s + (0.166 + 0.986i)16-s + (−0.261 − 0.261i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0763 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0763 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.988276 - 1.06683i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.988276 - 1.06683i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.486 + 1.32i)T \) |
| 3 | \( 1 + (-1.18 - 1.26i)T \) |
| good | 5 | \( 1 + (1.96 + 2.93i)T + (-1.91 + 4.61i)T^{2} \) |
| 7 | \( 1 + (-1.41 + 3.41i)T + (-4.94 - 4.94i)T^{2} \) |
| 11 | \( 1 + (-1.35 - 0.269i)T + (10.1 + 4.20i)T^{2} \) |
| 13 | \( 1 + (-3.69 - 2.47i)T + (4.97 + 12.0i)T^{2} \) |
| 17 | \( 1 + (1.07 + 1.07i)T + 17iT^{2} \) |
| 19 | \( 1 + (-2.55 - 1.70i)T + (7.27 + 17.5i)T^{2} \) |
| 23 | \( 1 + (0.348 + 0.842i)T + (-16.2 + 16.2i)T^{2} \) |
| 29 | \( 1 + (-1.64 - 8.26i)T + (-26.7 + 11.0i)T^{2} \) |
| 31 | \( 1 + 7.50T + 31T^{2} \) |
| 37 | \( 1 + (1.83 + 2.74i)T + (-14.1 + 34.1i)T^{2} \) |
| 41 | \( 1 + (-9.24 + 3.82i)T + (28.9 - 28.9i)T^{2} \) |
| 43 | \( 1 + (2.87 + 0.571i)T + (39.7 + 16.4i)T^{2} \) |
| 47 | \( 1 + (-3.91 + 3.91i)T - 47iT^{2} \) |
| 53 | \( 1 + (0.750 - 3.77i)T + (-48.9 - 20.2i)T^{2} \) |
| 59 | \( 1 + (-1.97 + 1.32i)T + (22.5 - 54.5i)T^{2} \) |
| 61 | \( 1 + (-2.15 - 10.8i)T + (-56.3 + 23.3i)T^{2} \) |
| 67 | \( 1 + (-1.15 + 0.230i)T + (61.8 - 25.6i)T^{2} \) |
| 71 | \( 1 + (7.50 + 3.10i)T + (50.2 + 50.2i)T^{2} \) |
| 73 | \( 1 + (3.51 - 1.45i)T + (51.6 - 51.6i)T^{2} \) |
| 79 | \( 1 + (-8.03 + 8.03i)T - 79iT^{2} \) |
| 83 | \( 1 + (-2.45 + 3.66i)T + (-31.7 - 76.6i)T^{2} \) |
| 89 | \( 1 + (-0.100 - 0.0416i)T + (62.9 + 62.9i)T^{2} \) |
| 97 | \( 1 - 12.5iT - 97T^{2} \) |
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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
−12.23380699847344173105134084484, −11.20639396708312809633858011926, −10.54115004422404776465725544660, −9.170725616946168940104322186350, −8.697294527263348247957614270025, −7.49226806708536042825021348792, −5.20685679936358632814388566956, −4.19382374788311336089722873961, −3.74427261398059375580675048785, −1.35315627472495104275201819177,
2.78646039280885511058227705650, 3.84203980269103850450380797829, 5.78350037524111340478982425068, 6.65659682533512578533291459125, 7.74496832431627693440308754846, 8.329850249449066896617642362057, 9.361462004401902763893476603691, 11.21197537267967748384669679883, 11.93699580314751248763028212286, 12.97264195438063921590528575828
