| L(s) = 1 | + (−1.51 − 1.10i)2-s + (−0.496 − 1.52i)3-s + (−1.38 − 4.26i)4-s + (2.00 − 10.9i)5-s + (−0.930 + 2.86i)6-s − 5.91·7-s + (−7.23 + 22.2i)8-s + (19.7 − 14.3i)9-s + (−15.1 + 14.4i)10-s + (46.2 + 33.5i)11-s + (−5.82 + 4.23i)12-s + (−23.8 + 17.3i)13-s + (8.97 + 6.52i)14-s + (−17.7 + 2.39i)15-s + (6.47 − 4.70i)16-s + (16.1 − 49.8i)17-s + ⋯ |
| L(s) = 1 | + (−0.536 − 0.389i)2-s + (−0.0955 − 0.293i)3-s + (−0.173 − 0.533i)4-s + (0.179 − 0.983i)5-s + (−0.0633 + 0.194i)6-s − 0.319·7-s + (−0.319 + 0.983i)8-s + (0.731 − 0.531i)9-s + (−0.479 + 0.457i)10-s + (1.26 + 0.920i)11-s + (−0.140 + 0.101i)12-s + (−0.509 + 0.370i)13-s + (0.171 + 0.124i)14-s + (−0.306 + 0.0412i)15-s + (0.101 − 0.0734i)16-s + (0.231 − 0.711i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.171 + 0.985i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.171 + 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.546948 - 0.650319i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.546948 - 0.650319i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-2.00 + 10.9i)T \) |
| good | 2 | \( 1 + (1.51 + 1.10i)T + (2.47 + 7.60i)T^{2} \) |
| 3 | \( 1 + (0.496 + 1.52i)T + (-21.8 + 15.8i)T^{2} \) |
| 7 | \( 1 + 5.91T + 343T^{2} \) |
| 11 | \( 1 + (-46.2 - 33.5i)T + (411. + 1.26e3i)T^{2} \) |
| 13 | \( 1 + (23.8 - 17.3i)T + (678. - 2.08e3i)T^{2} \) |
| 17 | \( 1 + (-16.1 + 49.8i)T + (-3.97e3 - 2.88e3i)T^{2} \) |
| 19 | \( 1 + (-28.8 + 88.8i)T + (-5.54e3 - 4.03e3i)T^{2} \) |
| 23 | \( 1 + (-101. - 73.5i)T + (3.75e3 + 1.15e4i)T^{2} \) |
| 29 | \( 1 + (-42.2 - 129. i)T + (-1.97e4 + 1.43e4i)T^{2} \) |
| 31 | \( 1 + (22.0 - 67.9i)T + (-2.41e4 - 1.75e4i)T^{2} \) |
| 37 | \( 1 + (149. - 108. i)T + (1.56e4 - 4.81e4i)T^{2} \) |
| 41 | \( 1 + (28.3 - 20.5i)T + (2.12e4 - 6.55e4i)T^{2} \) |
| 43 | \( 1 - 185.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (-130. - 401. i)T + (-8.39e4 + 6.10e4i)T^{2} \) |
| 53 | \( 1 + (214. + 659. i)T + (-1.20e5 + 8.75e4i)T^{2} \) |
| 59 | \( 1 + (466. - 338. i)T + (6.34e4 - 1.95e5i)T^{2} \) |
| 61 | \( 1 + (-29.5 - 21.4i)T + (7.01e4 + 2.15e5i)T^{2} \) |
| 67 | \( 1 + (48.1 - 148. i)T + (-2.43e5 - 1.76e5i)T^{2} \) |
| 71 | \( 1 + (-120. - 372. i)T + (-2.89e5 + 2.10e5i)T^{2} \) |
| 73 | \( 1 + (-946. - 687. i)T + (1.20e5 + 3.69e5i)T^{2} \) |
| 79 | \( 1 + (-323. - 995. i)T + (-3.98e5 + 2.89e5i)T^{2} \) |
| 83 | \( 1 + (-289. + 892. i)T + (-4.62e5 - 3.36e5i)T^{2} \) |
| 89 | \( 1 + (378. + 274. i)T + (2.17e5 + 6.70e5i)T^{2} \) |
| 97 | \( 1 + (529. + 1.62e3i)T + (-7.38e5 + 5.36e5i)T^{2} \) |
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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
−17.15413299428387081503181180486, −15.59201316776765654060968381272, −14.20159803458857090163602862372, −12.71809849732304621271659416765, −11.63861223758244071590681953732, −9.662304350093771979951249177460, −9.172924935652237305017420656416, −6.89356856271258870201393422288, −4.85842425751679227269797292188, −1.27046166393330651616269706009,
3.64930008310892366244504344131, 6.40095465363812078874802249446, 7.77370497713140782665076841383, 9.426164449941161072808951969467, 10.64205568158167560724413976951, 12.31802840915899243421085414056, 13.81196246018609724356267121778, 15.17082419121182364581066619829, 16.47754631702876012525149356415, 17.23852832631166879280919075284
