| L(s) = 1 | + (−0.991 + 1.36i)3-s + (−6.73 − 8.92i)5-s + 19.2i·7-s + (7.46 + 22.9i)9-s + (−10.1 + 31.1i)11-s + (−57.7 + 18.7i)13-s + (18.8 − 0.337i)15-s + (−26.1 − 35.9i)17-s + (−66.8 + 48.5i)19-s + (−26.3 − 19.1i)21-s + (71.5 + 23.2i)23-s + (−34.3 + 120. i)25-s + (−82.0 − 26.6i)27-s + (148. + 107. i)29-s + (218. − 158. i)31-s + ⋯ |
| L(s) = 1 | + (−0.190 + 0.262i)3-s + (−0.602 − 0.798i)5-s + 1.04i·7-s + (0.276 + 0.850i)9-s + (−0.277 + 0.853i)11-s + (−1.23 + 0.400i)13-s + (0.324 − 0.00580i)15-s + (−0.372 − 0.512i)17-s + (−0.807 + 0.586i)19-s + (−0.273 − 0.198i)21-s + (0.648 + 0.210i)23-s + (−0.274 + 0.961i)25-s + (−0.585 − 0.190i)27-s + (0.949 + 0.690i)29-s + (1.26 − 0.919i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.565 - 0.824i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.565 - 0.824i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.366637 + 0.696120i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.366637 + 0.696120i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (6.73 + 8.92i)T \) |
| good | 3 | \( 1 + (0.991 - 1.36i)T + (-8.34 - 25.6i)T^{2} \) |
| 7 | \( 1 - 19.2iT - 343T^{2} \) |
| 11 | \( 1 + (10.1 - 31.1i)T + (-1.07e3 - 782. i)T^{2} \) |
| 13 | \( 1 + (57.7 - 18.7i)T + (1.77e3 - 1.29e3i)T^{2} \) |
| 17 | \( 1 + (26.1 + 35.9i)T + (-1.51e3 + 4.67e3i)T^{2} \) |
| 19 | \( 1 + (66.8 - 48.5i)T + (2.11e3 - 6.52e3i)T^{2} \) |
| 23 | \( 1 + (-71.5 - 23.2i)T + (9.84e3 + 7.15e3i)T^{2} \) |
| 29 | \( 1 + (-148. - 107. i)T + (7.53e3 + 2.31e4i)T^{2} \) |
| 31 | \( 1 + (-218. + 158. i)T + (9.20e3 - 2.83e4i)T^{2} \) |
| 37 | \( 1 + (297. - 96.8i)T + (4.09e4 - 2.97e4i)T^{2} \) |
| 41 | \( 1 + (79.4 + 244. i)T + (-5.57e4 + 4.05e4i)T^{2} \) |
| 43 | \( 1 + 22.7iT - 7.95e4T^{2} \) |
| 47 | \( 1 + (-256. + 353. i)T + (-3.20e4 - 9.87e4i)T^{2} \) |
| 53 | \( 1 + (315. - 434. i)T + (-4.60e4 - 1.41e5i)T^{2} \) |
| 59 | \( 1 + (-186. - 574. i)T + (-1.66e5 + 1.20e5i)T^{2} \) |
| 61 | \( 1 + (-156. + 482. i)T + (-1.83e5 - 1.33e5i)T^{2} \) |
| 67 | \( 1 + (231. + 319. i)T + (-9.29e4 + 2.86e5i)T^{2} \) |
| 71 | \( 1 + (38.5 + 28.0i)T + (1.10e5 + 3.40e5i)T^{2} \) |
| 73 | \( 1 + (576. + 187. i)T + (3.14e5 + 2.28e5i)T^{2} \) |
| 79 | \( 1 + (-858. - 624. i)T + (1.52e5 + 4.68e5i)T^{2} \) |
| 83 | \( 1 + (-278. - 383. i)T + (-1.76e5 + 5.43e5i)T^{2} \) |
| 89 | \( 1 + (230. - 709. i)T + (-5.70e5 - 4.14e5i)T^{2} \) |
| 97 | \( 1 + (601. - 828. i)T + (-2.82e5 - 8.68e5i)T^{2} \) |
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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
−13.63888867636520951863704754834, −12.36893314906728904318378956016, −11.98272499402284895703214017060, −10.51542223584175728419389264016, −9.377346777379169705895533248577, −8.306999002141353941480136082632, −7.12613377544069153612300257275, −5.24929312268869027402258824721, −4.52029120771967528432813445488, −2.24654598384897534362183309170,
0.44488648775928711690004341908, 3.05435632354574058290553745238, 4.45921291444643477865307585786, 6.43669020744694739223948958362, 7.19903130574642700304045561498, 8.410979090096485280225423399935, 10.08397307258923398105834369769, 10.84334807560475573046334326464, 11.94151789959294461703550849418, 12.98307998395452061526014760739
