| 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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.174 + 0.984i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.174 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.01105 - 0.847894i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.01105 - 0.847894i\) |
| \(L(\frac{5}{2})\) |
|
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
−12.54690944453552461622589300080, −11.43646355545740333094175270622, −10.98904934127650641876137580555, −10.06979894488009134348609606361, −7.77889410339535600826926279244, −7.14732464492621625226472879584, −6.19913963325532758755891142101, −5.04551756793461835024778430917, −2.76905993487206565773767915916, −0.77426074864881540480666720259,
1.81972912169675328882669718481, 4.11752960177853714288471098436, 4.99415858454443034572909673071, 6.36455201400495249679735050913, 7.945198681768557281931484509310, 8.898966118435750907690751695704, 10.42716395808966095534017716927, 11.30921857418677787868094515085, 12.11221313159727427475408358319, 12.60217597955735619486346579674
