| L(s) = 1 | + (1.53 − 1.28i)2-s + (−2.66 − 2.23i)3-s + (0.694 − 3.93i)4-s + (−17.8 + 6.49i)5-s − 6.96·6-s + (−11.9 + 4.34i)7-s + (−4.00 − 6.92i)8-s + (−2.58 − 14.6i)9-s + (−18.9 + 32.8i)10-s + (8.70 + 15.0i)11-s + (−10.6 + 8.94i)12-s + (1.74 − 9.89i)13-s + (−12.7 + 22.0i)14-s + (62.1 + 22.6i)15-s + (−15.0 − 5.47i)16-s + (−3.12 − 17.6i)17-s + ⋯ |
| L(s) = 1 | + (0.541 − 0.454i)2-s + (−0.513 − 0.430i)3-s + (0.0868 − 0.492i)4-s + (−1.59 + 0.581i)5-s − 0.473·6-s + (−0.644 + 0.234i)7-s + (−0.176 − 0.306i)8-s + (−0.0957 − 0.542i)9-s + (−0.600 + 1.04i)10-s + (0.238 + 0.413i)11-s + (−0.256 + 0.215i)12-s + (0.0372 − 0.211i)13-s + (−0.242 + 0.420i)14-s + (1.06 + 0.389i)15-s + (−0.234 − 0.0855i)16-s + (−0.0445 − 0.252i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.969 - 0.243i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.969 - 0.243i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.0455375 + 0.367653i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0455375 + 0.367653i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.53 + 1.28i)T \) |
| 37 | \( 1 + (-78.2 + 211. i)T \) |
| good | 3 | \( 1 + (2.66 + 2.23i)T + (4.68 + 26.5i)T^{2} \) |
| 5 | \( 1 + (17.8 - 6.49i)T + (95.7 - 80.3i)T^{2} \) |
| 7 | \( 1 + (11.9 - 4.34i)T + (262. - 220. i)T^{2} \) |
| 11 | \( 1 + (-8.70 - 15.0i)T + (-665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 + (-1.74 + 9.89i)T + (-2.06e3 - 751. i)T^{2} \) |
| 17 | \( 1 + (3.12 + 17.6i)T + (-4.61e3 + 1.68e3i)T^{2} \) |
| 19 | \( 1 + (113. + 95.2i)T + (1.19e3 + 6.75e3i)T^{2} \) |
| 23 | \( 1 + (-19.8 + 34.3i)T + (-6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 + (-38.7 - 67.1i)T + (-1.21e4 + 2.11e4i)T^{2} \) |
| 31 | \( 1 - 23.0T + 2.97e4T^{2} \) |
| 41 | \( 1 + (55.2 - 313. i)T + (-6.47e4 - 2.35e4i)T^{2} \) |
| 43 | \( 1 + 232.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (-210. + 364. i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + (-398. - 144. i)T + (1.14e5 + 9.56e4i)T^{2} \) |
| 59 | \( 1 + (610. + 222. i)T + (1.57e5 + 1.32e5i)T^{2} \) |
| 61 | \( 1 + (52.4 - 297. i)T + (-2.13e5 - 7.76e4i)T^{2} \) |
| 67 | \( 1 + (754. - 274. i)T + (2.30e5 - 1.93e5i)T^{2} \) |
| 71 | \( 1 + (234. + 196. i)T + (6.21e4 + 3.52e5i)T^{2} \) |
| 73 | \( 1 + 994.T + 3.89e5T^{2} \) |
| 79 | \( 1 + (-698. + 254. i)T + (3.77e5 - 3.16e5i)T^{2} \) |
| 83 | \( 1 + (75.2 + 426. i)T + (-5.37e5 + 1.95e5i)T^{2} \) |
| 89 | \( 1 + (-1.11e3 - 404. i)T + (5.40e5 + 4.53e5i)T^{2} \) |
| 97 | \( 1 + (-912. + 1.58e3i)T + (-4.56e5 - 7.90e5i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.10576227926829664163709814206, −12.24691620192246585569717845113, −11.55197839892200399994310426595, −10.58684545673197465770877654452, −8.908573879231882028807321920776, −7.21378383016436064301779403792, −6.35097953753471075573727985582, −4.40542648573788829110322144185, −3.07242638802697097684359966706, −0.20735391891843347159433154263,
3.70986688170399526833437335038, 4.64097115201612665502590044119, 6.18256214682297151006053578739, 7.68430020911779784490216200021, 8.635568881004944406157403303304, 10.47292962590739090777706332983, 11.58013488387755907067533736103, 12.40868371365234448013454119419, 13.51362888736482264222359994744, 14.93892423426352536611604324030
