| L(s) = 1 | + (−0.226 − 0.696i)2-s + (−4.79 − 3.48i)3-s + (6.03 − 4.38i)4-s + (−3.97 + 12.2i)5-s + (−1.34 + 4.12i)6-s + (−13.6 + 9.95i)7-s + (−9.15 − 6.65i)8-s + (2.51 + 7.74i)9-s + 9.41·10-s − 44.2·12-s + (23.0 + 70.9i)13-s + (10.0 + 7.28i)14-s + (61.6 − 44.7i)15-s + (15.8 − 48.9i)16-s + (−25.5 + 78.7i)17-s + (4.82 − 3.50i)18-s + ⋯ |
| L(s) = 1 | + (−0.0799 − 0.246i)2-s + (−0.922 − 0.670i)3-s + (0.754 − 0.548i)4-s + (−0.355 + 1.09i)5-s + (−0.0912 + 0.280i)6-s + (−0.739 + 0.537i)7-s + (−0.404 − 0.294i)8-s + (0.0932 + 0.286i)9-s + 0.297·10-s − 1.06·12-s + (0.492 + 1.51i)13-s + (0.191 + 0.139i)14-s + (1.06 − 0.771i)15-s + (0.248 − 0.764i)16-s + (−0.364 + 1.12i)17-s + (0.0631 − 0.0458i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.263 - 0.964i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.263 - 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.563124 + 0.429951i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.563124 + 0.429951i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 \) |
| good | 2 | \( 1 + (0.226 + 0.696i)T + (-6.47 + 4.70i)T^{2} \) |
| 3 | \( 1 + (4.79 + 3.48i)T + (8.34 + 25.6i)T^{2} \) |
| 5 | \( 1 + (3.97 - 12.2i)T + (-101. - 73.4i)T^{2} \) |
| 7 | \( 1 + (13.6 - 9.95i)T + (105. - 326. i)T^{2} \) |
| 13 | \( 1 + (-23.0 - 70.9i)T + (-1.77e3 + 1.29e3i)T^{2} \) |
| 17 | \( 1 + (25.5 - 78.7i)T + (-3.97e3 - 2.88e3i)T^{2} \) |
| 19 | \( 1 + (-54.9 - 39.9i)T + (2.11e3 + 6.52e3i)T^{2} \) |
| 23 | \( 1 - 13.3T + 1.21e4T^{2} \) |
| 29 | \( 1 + (136. - 99.3i)T + (7.53e3 - 2.31e4i)T^{2} \) |
| 31 | \( 1 + (20.2 + 62.2i)T + (-2.41e4 + 1.75e4i)T^{2} \) |
| 37 | \( 1 + (33.0 - 24.0i)T + (1.56e4 - 4.81e4i)T^{2} \) |
| 41 | \( 1 + (222. + 161. i)T + (2.12e4 + 6.55e4i)T^{2} \) |
| 43 | \( 1 + 2.28T + 7.95e4T^{2} \) |
| 47 | \( 1 + (58.1 + 42.2i)T + (3.20e4 + 9.87e4i)T^{2} \) |
| 53 | \( 1 + (46.0 + 141. i)T + (-1.20e5 + 8.75e4i)T^{2} \) |
| 59 | \( 1 + (441. - 320. i)T + (6.34e4 - 1.95e5i)T^{2} \) |
| 61 | \( 1 + (-31.3 + 96.3i)T + (-1.83e5 - 1.33e5i)T^{2} \) |
| 67 | \( 1 - 411.T + 3.00e5T^{2} \) |
| 71 | \( 1 + (145. - 447. i)T + (-2.89e5 - 2.10e5i)T^{2} \) |
| 73 | \( 1 + (493. - 358. i)T + (1.20e5 - 3.69e5i)T^{2} \) |
| 79 | \( 1 + (302. + 930. i)T + (-3.98e5 + 2.89e5i)T^{2} \) |
| 83 | \( 1 + (-8.08 + 24.8i)T + (-4.62e5 - 3.36e5i)T^{2} \) |
| 89 | \( 1 + 352.T + 7.04e5T^{2} \) |
| 97 | \( 1 + (-261. - 806. i)T + (-7.38e5 + 5.36e5i)T^{2} \) |
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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
−12.86610121325427754466146687604, −11.84217775698184350841442467059, −11.31858035081209135926322767350, −10.43237265475413915695617722032, −9.155470441107090568258038759372, −7.19141991350041409376732414077, −6.52608652902308103358975789572, −5.82200617313300127203682278126, −3.42118731669303917393884140558, −1.74658442665555663915037583988,
0.41413139738033480955694756895, 3.28365430240730959046458589908, 4.80941548048462734144191655125, 5.88594266144103788394554571030, 7.26124891607668762523466376458, 8.367258175845034565244324794352, 9.708945614947093901915641678612, 10.85662498265275840450669332782, 11.64507895992756581641445881947, 12.64835562386402282279096000756
