L(s) = 1 | + (−8.14 − 4.70i)2-s + (28.2 + 48.8i)4-s + (33.1 − 57.4i)5-s + (−116. + 57.3i)7-s − 229. i·8-s + (−540. + 312. i)10-s + (−383. + 221. i)11-s + 437. i·13-s + (1.21e3 + 79.6i)14-s + (−176. + 306. i)16-s + (−476. − 825. i)17-s + (1.93e3 + 1.11e3i)19-s + 3.74e3·20-s + 4.16e3·22-s + (2.09e3 + 1.21e3i)23-s + ⋯ |
L(s) = 1 | + (−1.43 − 0.831i)2-s + (0.881 + 1.52i)4-s + (0.593 − 1.02i)5-s + (−0.896 + 0.442i)7-s − 1.26i·8-s + (−1.70 + 0.986i)10-s + (−0.955 + 0.551i)11-s + 0.717i·13-s + (1.65 + 0.108i)14-s + (−0.172 + 0.299i)16-s + (−0.399 − 0.692i)17-s + (1.22 + 0.709i)19-s + 2.09·20-s + 1.83·22-s + (0.826 + 0.477i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.911 - 0.410i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.911 - 0.410i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(0.550995 + 0.118392i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.550995 + 0.118392i\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (116. - 57.3i)T \) |
good | 2 | \( 1 + (8.14 + 4.70i)T + (16 + 27.7i)T^{2} \) |
| 5 | \( 1 + (-33.1 + 57.4i)T + (-1.56e3 - 2.70e3i)T^{2} \) |
| 11 | \( 1 + (383. - 221. i)T + (8.05e4 - 1.39e5i)T^{2} \) |
| 13 | \( 1 - 437. iT - 3.71e5T^{2} \) |
| 17 | \( 1 + (476. + 825. i)T + (-7.09e5 + 1.22e6i)T^{2} \) |
| 19 | \( 1 + (-1.93e3 - 1.11e3i)T + (1.23e6 + 2.14e6i)T^{2} \) |
| 23 | \( 1 + (-2.09e3 - 1.21e3i)T + (3.21e6 + 5.57e6i)T^{2} \) |
| 29 | \( 1 - 6.38e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 + (-5.91e3 + 3.41e3i)T + (1.43e7 - 2.47e7i)T^{2} \) |
| 37 | \( 1 + (7.60e3 - 1.31e4i)T + (-3.46e7 - 6.00e7i)T^{2} \) |
| 41 | \( 1 + 1.72e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.37e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + (-9.07e3 + 1.57e4i)T + (-1.14e8 - 1.98e8i)T^{2} \) |
| 53 | \( 1 + (2.07e4 - 1.20e4i)T + (2.09e8 - 3.62e8i)T^{2} \) |
| 59 | \( 1 + (-2.12e4 - 3.67e4i)T + (-3.57e8 + 6.19e8i)T^{2} \) |
| 61 | \( 1 + (1.77e4 + 1.02e4i)T + (4.22e8 + 7.31e8i)T^{2} \) |
| 67 | \( 1 + (290. + 503. i)T + (-6.75e8 + 1.16e9i)T^{2} \) |
| 71 | \( 1 - 3.26e4iT - 1.80e9T^{2} \) |
| 73 | \( 1 + (4.74e4 - 2.73e4i)T + (1.03e9 - 1.79e9i)T^{2} \) |
| 79 | \( 1 + (1.85e4 - 3.22e4i)T + (-1.53e9 - 2.66e9i)T^{2} \) |
| 83 | \( 1 + 4.46e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + (-1.20e4 + 2.09e4i)T + (-2.79e9 - 4.83e9i)T^{2} \) |
| 97 | \( 1 - 2.89e4iT - 8.58e9T^{2} \) |
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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
−13.62588915613647579896248671258, −12.58271152812529163764856785804, −11.70811502509274091719919149964, −10.19263433675177282728271865505, −9.447621938330221371235494982588, −8.697210701235581703565048549929, −7.21548210239666547250581042161, −5.25225157442924303283487723560, −2.79522705724784153549084617338, −1.26915683261486615390277835769,
0.46143508320704541986290625631, 2.86236506612957415798512362495, 5.85150380611362600938401588958, 6.81721013977859308346619370925, 7.84532410428670353562415926658, 9.246399694137609014856863158908, 10.30593000441801220258838009876, 10.81685523604123532839789644432, 13.01776400760230001613375433686, 14.12934336169496032236750637397