| L(s) = 1 | + (−0.840 − 2.58i)2-s + (−6.97 − 5.06i)3-s + (0.486 − 0.353i)4-s + (6.29 − 19.3i)5-s + (−7.24 + 22.2i)6-s + (5.66 − 4.11i)7-s + (−18.9 − 13.7i)8-s + (14.6 + 44.9i)9-s − 55.3·10-s + (23.1 + 28.1i)11-s − 5.18·12-s + (4.09 + 12.5i)13-s + (−15.4 − 11.1i)14-s + (−141. + 103. i)15-s + (−18.1 + 55.9i)16-s + (27.2 − 83.7i)17-s + ⋯ |
| L(s) = 1 | + (−0.297 − 0.914i)2-s + (−1.34 − 0.975i)3-s + (0.0608 − 0.0442i)4-s + (0.562 − 1.73i)5-s + (−0.492 + 1.51i)6-s + (0.305 − 0.222i)7-s + (−0.836 − 0.607i)8-s + (0.541 + 1.66i)9-s − 1.75·10-s + (0.635 + 0.772i)11-s − 0.124·12-s + (0.0873 + 0.268i)13-s + (−0.294 − 0.213i)14-s + (−2.44 + 1.77i)15-s + (−0.284 + 0.874i)16-s + (0.388 − 1.19i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.817 - 0.576i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.817 - 0.576i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.287113 + 0.905574i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.287113 + 0.905574i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (-5.66 + 4.11i)T \) |
| 11 | \( 1 + (-23.1 - 28.1i)T \) |
| good | 2 | \( 1 + (0.840 + 2.58i)T + (-6.47 + 4.70i)T^{2} \) |
| 3 | \( 1 + (6.97 + 5.06i)T + (8.34 + 25.6i)T^{2} \) |
| 5 | \( 1 + (-6.29 + 19.3i)T + (-101. - 73.4i)T^{2} \) |
| 13 | \( 1 + (-4.09 - 12.5i)T + (-1.77e3 + 1.29e3i)T^{2} \) |
| 17 | \( 1 + (-27.2 + 83.7i)T + (-3.97e3 - 2.88e3i)T^{2} \) |
| 19 | \( 1 + (-84.7 - 61.5i)T + (2.11e3 + 6.52e3i)T^{2} \) |
| 23 | \( 1 - 79.6T + 1.21e4T^{2} \) |
| 29 | \( 1 + (-59.5 + 43.2i)T + (7.53e3 - 2.31e4i)T^{2} \) |
| 31 | \( 1 + (23.0 + 70.9i)T + (-2.41e4 + 1.75e4i)T^{2} \) |
| 37 | \( 1 + (282. - 205. i)T + (1.56e4 - 4.81e4i)T^{2} \) |
| 41 | \( 1 + (23.1 + 16.7i)T + (2.12e4 + 6.55e4i)T^{2} \) |
| 43 | \( 1 - 293.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (-187. - 136. i)T + (3.20e4 + 9.87e4i)T^{2} \) |
| 53 | \( 1 + (32.3 + 99.5i)T + (-1.20e5 + 8.75e4i)T^{2} \) |
| 59 | \( 1 + (234. - 170. i)T + (6.34e4 - 1.95e5i)T^{2} \) |
| 61 | \( 1 + (-107. + 332. i)T + (-1.83e5 - 1.33e5i)T^{2} \) |
| 67 | \( 1 + 882.T + 3.00e5T^{2} \) |
| 71 | \( 1 + (123. - 378. i)T + (-2.89e5 - 2.10e5i)T^{2} \) |
| 73 | \( 1 + (289. - 210. i)T + (1.20e5 - 3.69e5i)T^{2} \) |
| 79 | \( 1 + (0.463 + 1.42i)T + (-3.98e5 + 2.89e5i)T^{2} \) |
| 83 | \( 1 + (-170. + 525. i)T + (-4.62e5 - 3.36e5i)T^{2} \) |
| 89 | \( 1 - 398.T + 7.04e5T^{2} \) |
| 97 | \( 1 + (115. + 356. i)T + (-7.38e5 + 5.36e5i)T^{2} \) |
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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
−12.76627820563004373796273706779, −12.05316363770872466596540136029, −11.58877361424573510381355567744, −10.09049842665745388319036289537, −9.107272160863749085234477292048, −7.30856325109112511227371932481, −5.92005090283863754296723343109, −4.84329279877125532504616774627, −1.61902070726149973374199283063, −0.850749978875430655735668307170,
3.29989062812691108136668251623, 5.54666467631204101252740006679, 6.23180003376861993723641026897, 7.23281340777242430991903424953, 9.075973819945444825799282122012, 10.48718674246044932311220498146, 11.05447781458187607151752031282, 11.98133248181547904369511482204, 14.09637108564139138392430451251, 14.98749357571976254683273100919
