| L(s) = 1 | + (1.87 + 0.684i)2-s + (−4.40 + 1.60i)3-s + (3.06 + 2.57i)4-s + (−0.0856 + 0.485i)5-s − 9.36·6-s + (−4.85 + 27.5i)7-s + (4.00 + 6.92i)8-s + (−3.88 + 3.26i)9-s + (−0.493 + 0.854i)10-s + (14.3 + 24.8i)11-s + (−17.6 − 6.40i)12-s + (−32.8 − 27.5i)13-s + (−27.9 + 48.4i)14-s + (−0.401 − 2.27i)15-s + (2.77 + 15.7i)16-s + (−34.3 + 28.7i)17-s + ⋯ |
| L(s) = 1 | + (0.664 + 0.241i)2-s + (−0.846 + 0.308i)3-s + (0.383 + 0.321i)4-s + (−0.00766 + 0.0434i)5-s − 0.637·6-s + (−0.262 + 1.48i)7-s + (0.176 + 0.306i)8-s + (−0.143 + 0.120i)9-s + (−0.0155 + 0.0270i)10-s + (0.392 + 0.680i)11-s + (−0.423 − 0.154i)12-s + (−0.700 − 0.587i)13-s + (−0.533 + 0.924i)14-s + (−0.00690 − 0.0391i)15-s + (0.0434 + 0.246i)16-s + (−0.489 + 0.410i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.405 - 0.914i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.405 - 0.914i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.765705 + 1.17664i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.765705 + 1.17664i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.87 - 0.684i)T \) |
| 37 | \( 1 + (74.8 - 212. i)T \) |
| good | 3 | \( 1 + (4.40 - 1.60i)T + (20.6 - 17.3i)T^{2} \) |
| 5 | \( 1 + (0.0856 - 0.485i)T + (-117. - 42.7i)T^{2} \) |
| 7 | \( 1 + (4.85 - 27.5i)T + (-322. - 117. i)T^{2} \) |
| 11 | \( 1 + (-14.3 - 24.8i)T + (-665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 + (32.8 + 27.5i)T + (381. + 2.16e3i)T^{2} \) |
| 17 | \( 1 + (34.3 - 28.7i)T + (853. - 4.83e3i)T^{2} \) |
| 19 | \( 1 + (-146. + 53.2i)T + (5.25e3 - 4.40e3i)T^{2} \) |
| 23 | \( 1 + (-17.3 + 29.9i)T + (-6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 + (4.83 + 8.36i)T + (-1.21e4 + 2.11e4i)T^{2} \) |
| 31 | \( 1 - 9.17T + 2.97e4T^{2} \) |
| 41 | \( 1 + (7.15 + 6.00i)T + (1.19e4 + 6.78e4i)T^{2} \) |
| 43 | \( 1 + 87.8T + 7.95e4T^{2} \) |
| 47 | \( 1 + (-125. + 217. i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + (77.0 + 436. i)T + (-1.39e5 + 5.09e4i)T^{2} \) |
| 59 | \( 1 + (-136. - 774. i)T + (-1.92e5 + 7.02e4i)T^{2} \) |
| 61 | \( 1 + (-598. - 502. i)T + (3.94e4 + 2.23e5i)T^{2} \) |
| 67 | \( 1 + (122. - 696. i)T + (-2.82e5 - 1.02e5i)T^{2} \) |
| 71 | \( 1 + (-156. + 56.7i)T + (2.74e5 - 2.30e5i)T^{2} \) |
| 73 | \( 1 + 427.T + 3.89e5T^{2} \) |
| 79 | \( 1 + (87.9 - 498. i)T + (-4.63e5 - 1.68e5i)T^{2} \) |
| 83 | \( 1 + (-824. + 691. i)T + (9.92e4 - 5.63e5i)T^{2} \) |
| 89 | \( 1 + (134. + 760. i)T + (-6.62e5 + 2.41e5i)T^{2} \) |
| 97 | \( 1 + (-100. + 174. i)T + (-4.56e5 - 7.90e5i)T^{2} \) |
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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
−14.67291492132173633793307115095, −13.17770086757739348906959185336, −12.08858427274518141198247144714, −11.54103122541697524359882007988, −10.07105884518408196222833162325, −8.711832043004412280351992791558, −7.00979180160965261504565755041, −5.66645743589733086759221571594, −4.91747799209968354612134779500, −2.75754129568096105091209037485,
0.840856993379625721794432695435, 3.51195621462715063024924423445, 5.03485426697783314665404092744, 6.44944577003903660760991135587, 7.37373128846545847444121260724, 9.464862673224511629155478218467, 10.77393515554528516753188873654, 11.58653868659103274334961867421, 12.54392053020281732809457073633, 13.81404922883918683628538430464
