| L(s) = 1 | + (0.618 − 1.90i)2-s + (−3.92 + 2.85i)3-s + (−3.23 − 2.35i)4-s + (−1.54 − 4.75i)5-s + (3.00 + 9.23i)6-s + (5.32 + 3.86i)7-s + (−6.47 + 4.70i)8-s + (−1.05 + 3.25i)9-s − 10.0·10-s + (18.0 + 31.7i)11-s + 19.4·12-s + (−18.7 + 57.6i)13-s + (10.6 − 7.73i)14-s + (19.6 + 14.2i)15-s + (4.94 + 15.2i)16-s + (26.5 + 81.8i)17-s + ⋯ |
| L(s) = 1 | + (0.218 − 0.672i)2-s + (−0.755 + 0.549i)3-s + (−0.404 − 0.293i)4-s + (−0.138 − 0.425i)5-s + (0.204 + 0.628i)6-s + (0.287 + 0.208i)7-s + (−0.286 + 0.207i)8-s + (−0.0391 + 0.120i)9-s − 0.316·10-s + (0.493 + 0.869i)11-s + 0.467·12-s + (−0.399 + 1.22i)13-s + (0.203 − 0.147i)14-s + (0.338 + 0.245i)15-s + (0.0772 + 0.237i)16-s + (0.379 + 1.16i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.371 - 0.928i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 110 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.371 - 0.928i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.794833 + 0.538071i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.794833 + 0.538071i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.618 + 1.90i)T \) |
| 5 | \( 1 + (1.54 + 4.75i)T \) |
| 11 | \( 1 + (-18.0 - 31.7i)T \) |
| good | 3 | \( 1 + (3.92 - 2.85i)T + (8.34 - 25.6i)T^{2} \) |
| 7 | \( 1 + (-5.32 - 3.86i)T + (105. + 326. i)T^{2} \) |
| 13 | \( 1 + (18.7 - 57.6i)T + (-1.77e3 - 1.29e3i)T^{2} \) |
| 17 | \( 1 + (-26.5 - 81.8i)T + (-3.97e3 + 2.88e3i)T^{2} \) |
| 19 | \( 1 + (-2.21 + 1.60i)T + (2.11e3 - 6.52e3i)T^{2} \) |
| 23 | \( 1 + 70.7T + 1.21e4T^{2} \) |
| 29 | \( 1 + (2.67 + 1.94i)T + (7.53e3 + 2.31e4i)T^{2} \) |
| 31 | \( 1 + (-30.9 + 95.2i)T + (-2.41e4 - 1.75e4i)T^{2} \) |
| 37 | \( 1 + (-264. - 191. i)T + (1.56e4 + 4.81e4i)T^{2} \) |
| 41 | \( 1 + (16.6 - 12.1i)T + (2.12e4 - 6.55e4i)T^{2} \) |
| 43 | \( 1 + 469.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (238. - 173. i)T + (3.20e4 - 9.87e4i)T^{2} \) |
| 53 | \( 1 + (121. - 373. i)T + (-1.20e5 - 8.75e4i)T^{2} \) |
| 59 | \( 1 + (-477. - 347. i)T + (6.34e4 + 1.95e5i)T^{2} \) |
| 61 | \( 1 + (70.0 + 215. i)T + (-1.83e5 + 1.33e5i)T^{2} \) |
| 67 | \( 1 + 177.T + 3.00e5T^{2} \) |
| 71 | \( 1 + (163. + 504. i)T + (-2.89e5 + 2.10e5i)T^{2} \) |
| 73 | \( 1 + (-734. - 533. i)T + (1.20e5 + 3.69e5i)T^{2} \) |
| 79 | \( 1 + (-260. + 802. i)T + (-3.98e5 - 2.89e5i)T^{2} \) |
| 83 | \( 1 + (-149. - 460. i)T + (-4.62e5 + 3.36e5i)T^{2} \) |
| 89 | \( 1 + 60.8T + 7.04e5T^{2} \) |
| 97 | \( 1 + (-79.2 + 244. i)T + (-7.38e5 - 5.36e5i)T^{2} \) |
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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
−13.15134169803468967873329205906, −11.97194568006056436837335815168, −11.54603408754412595480618450857, −10.28252765980396979104157359714, −9.465403520533077540722165733843, −8.094629874497301186882541504435, −6.30155745778655762512825432089, −4.93545913443420567199605617826, −4.13016955139014695709850161162, −1.83811215170196671675891062311,
0.54949703419263031430273600822, 3.33692212565432408276811844161, 5.18690494789197914724195961667, 6.20268149792233739495497777644, 7.22286596416538320112581942853, 8.265416626337633058295149022305, 9.778579625802917681215842340039, 11.16906217788216769574632096699, 11.96107013938567183466456297510, 13.01693109184223849865042057452
