| L(s) = 1 | + (−1.69 − 2.33i)2-s + (−1.33 + 4.10i)4-s + (−0.356 − 0.259i)5-s + (−10.0 − 3.27i)7-s + (0.881 − 0.286i)8-s + 1.27i·10-s + (−9.39 + 5.72i)11-s + (3.78 + 5.21i)13-s + (9.43 + 29.0i)14-s + (11.8 + 8.58i)16-s + (11.1 − 15.2i)17-s + (−26.1 + 8.51i)19-s + (1.54 − 1.12i)20-s + (29.2 + 12.2i)22-s − 6.29·23-s + ⋯ |
| L(s) = 1 | + (−0.847 − 1.16i)2-s + (−0.333 + 1.02i)4-s + (−0.0713 − 0.0518i)5-s + (−1.43 − 0.467i)7-s + (0.110 − 0.0358i)8-s + 0.127i·10-s + (−0.853 + 0.520i)11-s + (0.291 + 0.401i)13-s + (0.673 + 2.07i)14-s + (0.738 + 0.536i)16-s + (0.653 − 0.899i)17-s + (−1.37 + 0.447i)19-s + (0.0771 − 0.0560i)20-s + (1.33 + 0.555i)22-s − 0.273·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.578 - 0.815i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.578 - 0.815i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.0801604 + 0.155124i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0801604 + 0.155124i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 + (9.39 - 5.72i)T \) |
| good | 2 | \( 1 + (1.69 + 2.33i)T + (-1.23 + 3.80i)T^{2} \) |
| 5 | \( 1 + (0.356 + 0.259i)T + (7.72 + 23.7i)T^{2} \) |
| 7 | \( 1 + (10.0 + 3.27i)T + (39.6 + 28.8i)T^{2} \) |
| 13 | \( 1 + (-3.78 - 5.21i)T + (-52.2 + 160. i)T^{2} \) |
| 17 | \( 1 + (-11.1 + 15.2i)T + (-89.3 - 274. i)T^{2} \) |
| 19 | \( 1 + (26.1 - 8.51i)T + (292. - 212. i)T^{2} \) |
| 23 | \( 1 + 6.29T + 529T^{2} \) |
| 29 | \( 1 + (42.0 + 13.6i)T + (680. + 494. i)T^{2} \) |
| 31 | \( 1 + (-21.9 + 15.9i)T + (296. - 913. i)T^{2} \) |
| 37 | \( 1 + (0.263 - 0.811i)T + (-1.10e3 - 804. i)T^{2} \) |
| 41 | \( 1 + (-12.5 + 4.08i)T + (1.35e3 - 988. i)T^{2} \) |
| 43 | \( 1 + 68.8iT - 1.84e3T^{2} \) |
| 47 | \( 1 + (-4.98 - 15.3i)T + (-1.78e3 + 1.29e3i)T^{2} \) |
| 53 | \( 1 + (-31.2 + 22.7i)T + (868. - 2.67e3i)T^{2} \) |
| 59 | \( 1 + (23.7 - 73.1i)T + (-2.81e3 - 2.04e3i)T^{2} \) |
| 61 | \( 1 + (16.8 - 23.1i)T + (-1.14e3 - 3.53e3i)T^{2} \) |
| 67 | \( 1 + 78.0T + 4.48e3T^{2} \) |
| 71 | \( 1 + (21.9 + 15.9i)T + (1.55e3 + 4.79e3i)T^{2} \) |
| 73 | \( 1 + (12.3 + 4.01i)T + (4.31e3 + 3.13e3i)T^{2} \) |
| 79 | \( 1 + (-55.4 - 76.2i)T + (-1.92e3 + 5.93e3i)T^{2} \) |
| 83 | \( 1 + (-68.1 + 93.8i)T + (-2.12e3 - 6.55e3i)T^{2} \) |
| 89 | \( 1 + 65.2T + 7.92e3T^{2} \) |
| 97 | \( 1 + (-56.8 + 41.3i)T + (2.90e3 - 8.94e3i)T^{2} \) |
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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
−12.68817297747151023616620600298, −11.84328431960421702375154549360, −10.48516163087233487511940374424, −9.976159862962987045450530954556, −8.970771401411617395330469752347, −7.59126748890959922539961248662, −6.08940743055596444791442179135, −3.89000501979450366824285874930, −2.43345699889613030239758531372, −0.16332141490363799536104001133,
3.24935061189363381644476993180, 5.69342252495040536278374765195, 6.43669391155416646901253586459, 7.74423059728830306995451201477, 8.726573248595166819711482354162, 9.704984456525695586422780427422, 10.76731294144161056606046616888, 12.52699688068134846013197825748, 13.23214845566420543477066631791, 14.86708754873928690285026660060
