| L(s) = 1 | + (1.05 + 1.74i)2-s + (2.53 − 1.60i)3-s + (−0.0720 + 0.135i)4-s + (−0.320 − 2.94i)5-s + (5.46 + 2.75i)6-s + (−3.10 + 1.04i)7-s + (7.82 − 0.424i)8-s + (3.87 − 8.12i)9-s + (4.81 − 3.65i)10-s + (−4.65 − 3.15i)11-s + (0.0346 + 0.460i)12-s + (16.6 + 15.7i)13-s + (−5.08 − 4.31i)14-s + (−5.53 − 6.96i)15-s + (9.31 + 13.7i)16-s + (−0.371 + 1.10i)17-s + ⋯ |
| L(s) = 1 | + (0.525 + 0.873i)2-s + (0.845 − 0.533i)3-s + (−0.0180 + 0.0339i)4-s + (−0.0641 − 0.589i)5-s + (0.910 + 0.458i)6-s + (−0.442 + 0.149i)7-s + (0.978 − 0.0530i)8-s + (0.430 − 0.902i)9-s + (0.481 − 0.365i)10-s + (−0.422 − 0.286i)11-s + (0.00288 + 0.0383i)12-s + (1.28 + 1.21i)13-s + (−0.362 − 0.308i)14-s + (−0.368 − 0.464i)15-s + (0.581 + 0.858i)16-s + (−0.0218 + 0.0648i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 - 0.0822i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.996 - 0.0822i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.60068 + 0.107187i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.60068 + 0.107187i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-2.53 + 1.60i)T \) |
| 59 | \( 1 + (9.00 - 58.3i)T \) |
| good | 2 | \( 1 + (-1.05 - 1.74i)T + (-1.87 + 3.53i)T^{2} \) |
| 5 | \( 1 + (0.320 + 2.94i)T + (-24.4 + 5.37i)T^{2} \) |
| 7 | \( 1 + (3.10 - 1.04i)T + (39.0 - 29.6i)T^{2} \) |
| 11 | \( 1 + (4.65 + 3.15i)T + (44.7 + 112. i)T^{2} \) |
| 13 | \( 1 + (-16.6 - 15.7i)T + (9.14 + 168. i)T^{2} \) |
| 17 | \( 1 + (0.371 - 1.10i)T + (-230. - 174. i)T^{2} \) |
| 19 | \( 1 + (3.91 + 23.8i)T + (-342. + 115. i)T^{2} \) |
| 23 | \( 1 + (33.8 - 9.40i)T + (453. - 272. i)T^{2} \) |
| 29 | \( 1 + (17.3 - 28.7i)T + (-393. - 743. i)T^{2} \) |
| 31 | \( 1 + (9.22 - 56.2i)T + (-910. - 306. i)T^{2} \) |
| 37 | \( 1 + (2.70 - 49.9i)T + (-1.36e3 - 148. i)T^{2} \) |
| 41 | \( 1 + (58.0 + 16.1i)T + (1.44e3 + 866. i)T^{2} \) |
| 43 | \( 1 + (0.712 + 1.05i)T + (-684. + 1.71e3i)T^{2} \) |
| 47 | \( 1 + (-3.92 + 36.0i)T + (-2.15e3 - 474. i)T^{2} \) |
| 53 | \( 1 + (-48.5 + 63.8i)T + (-751. - 2.70e3i)T^{2} \) |
| 61 | \( 1 + (-21.4 + 12.9i)T + (1.74e3 - 3.28e3i)T^{2} \) |
| 67 | \( 1 + (3.09 + 57.1i)T + (-4.46e3 + 485. i)T^{2} \) |
| 71 | \( 1 + (-2.26 + 20.8i)T + (-4.92e3 - 1.08e3i)T^{2} \) |
| 73 | \( 1 + (10.6 - 12.5i)T + (-862. - 5.25e3i)T^{2} \) |
| 79 | \( 1 + (-16.9 + 42.5i)T + (-4.53e3 - 4.29e3i)T^{2} \) |
| 83 | \( 1 + (9.51 + 20.5i)T + (-4.45e3 + 5.25e3i)T^{2} \) |
| 89 | \( 1 + (7.41 - 12.3i)T + (-3.71e3 - 6.99e3i)T^{2} \) |
| 97 | \( 1 + (-75.2 - 88.6i)T + (-1.52e3 + 9.28e3i)T^{2} \) |
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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
−12.96496903103011346408385209320, −11.74725509258097148501097317450, −10.39819578075735309596813578602, −8.991926950523882543015616605439, −8.381169484721541479789013667066, −7.00680513397408266366936476824, −6.37141208493730306187995895657, −4.96587938263501306454861151701, −3.58964025253434341179851290824, −1.61527076390310073256778996093,
2.20556004094006544417354678165, 3.39685957442521680498482162833, 4.08796187729150644238907342860, 5.84310873182640763416222628242, 7.57149669455837689613139064375, 8.269002557505028101002433700734, 9.915811699775226333069838054162, 10.45046111706578714683870886884, 11.32118074711589320216287667158, 12.68041403432435147924286471809
