| L(s) = 1 | + (1.53 + 1.28i)2-s + (6.50 − 5.45i)3-s + (0.694 + 3.93i)4-s + (−8.13 − 2.95i)5-s + 16.9·6-s + (24.8 + 9.03i)7-s + (−4.00 + 6.92i)8-s + (7.82 − 44.3i)9-s + (−8.65 − 14.9i)10-s + (−5.47 + 9.48i)11-s + (26.0 + 21.8i)12-s + (−12.0 − 68.0i)13-s + (26.4 + 45.7i)14-s + (−69.0 + 25.1i)15-s + (−15.0 + 5.47i)16-s + (−14.0 + 79.7i)17-s + ⋯ |
| L(s) = 1 | + (0.541 + 0.454i)2-s + (1.25 − 1.05i)3-s + (0.0868 + 0.492i)4-s + (−0.727 − 0.264i)5-s + 1.15·6-s + (1.33 + 0.487i)7-s + (−0.176 + 0.306i)8-s + (0.289 − 1.64i)9-s + (−0.273 − 0.473i)10-s + (−0.150 + 0.259i)11-s + (0.625 + 0.525i)12-s + (−0.256 − 1.45i)13-s + (0.504 + 0.873i)14-s + (−1.18 + 0.432i)15-s + (−0.234 + 0.0855i)16-s + (−0.200 + 1.13i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.981 + 0.192i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.981 + 0.192i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.63282 - 0.256364i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.63282 - 0.256364i\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 + (-19.2 + 224. i)T \) |
| good | 3 | \( 1 + (-6.50 + 5.45i)T + (4.68 - 26.5i)T^{2} \) |
| 5 | \( 1 + (8.13 + 2.95i)T + (95.7 + 80.3i)T^{2} \) |
| 7 | \( 1 + (-24.8 - 9.03i)T + (262. + 220. i)T^{2} \) |
| 11 | \( 1 + (5.47 - 9.48i)T + (-665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 + (12.0 + 68.0i)T + (-2.06e3 + 751. i)T^{2} \) |
| 17 | \( 1 + (14.0 - 79.7i)T + (-4.61e3 - 1.68e3i)T^{2} \) |
| 19 | \( 1 + (68.8 - 57.7i)T + (1.19e3 - 6.75e3i)T^{2} \) |
| 23 | \( 1 + (-56.0 - 97.0i)T + (-6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 + (119. - 206. i)T + (-1.21e4 - 2.11e4i)T^{2} \) |
| 31 | \( 1 + 166.T + 2.97e4T^{2} \) |
| 41 | \( 1 + (42.7 + 242. i)T + (-6.47e4 + 2.35e4i)T^{2} \) |
| 43 | \( 1 - 406.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (-2.23 - 3.87i)T + (-5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 + (49.5 - 18.0i)T + (1.14e5 - 9.56e4i)T^{2} \) |
| 59 | \( 1 + (-405. + 147. i)T + (1.57e5 - 1.32e5i)T^{2} \) |
| 61 | \( 1 + (107. + 610. i)T + (-2.13e5 + 7.76e4i)T^{2} \) |
| 67 | \( 1 + (-681. - 248. i)T + (2.30e5 + 1.93e5i)T^{2} \) |
| 71 | \( 1 + (-343. + 288. i)T + (6.21e4 - 3.52e5i)T^{2} \) |
| 73 | \( 1 - 23.6T + 3.89e5T^{2} \) |
| 79 | \( 1 + (952. + 346. i)T + (3.77e5 + 3.16e5i)T^{2} \) |
| 83 | \( 1 + (-4.54 + 25.7i)T + (-5.37e5 - 1.95e5i)T^{2} \) |
| 89 | \( 1 + (742. - 270. i)T + (5.40e5 - 4.53e5i)T^{2} \) |
| 97 | \( 1 + (-775. - 1.34e3i)T + (-4.56e5 + 7.90e5i)T^{2} \) |
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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
−14.29308288270139217653728195195, −12.76775430413169172649410688889, −12.55498024483013735093977911425, −10.99191787144714603271129648759, −8.759624404087464037452810603868, −8.023939585804077933652364464322, −7.41349176771138482341063326430, −5.49751081045871219941470925811, −3.72668237294372909752736865191, −1.96975167753307247358981884102,
2.42640544482682178427133412525, 4.06653254211314527337805187256, 4.68543861501390778883175246288, 7.28811485857168972335338105578, 8.530613110514565889638676894863, 9.594273908558015375972560441884, 11.00750864947399219595902520547, 11.52547044926415241489592623564, 13.43026120750300976871444026879, 14.31596190753339538805222087788
