| L(s) = 1 | + (−1.87 − 0.684i)2-s + (−0.520 − 2.95i)3-s + (3.06 + 2.57i)4-s + (−0.233 + 0.195i)5-s + (−1.04 + 5.90i)6-s + (−9.88 + 17.1i)7-s + (−4.00 − 6.92i)8-s + (−8.45 + 3.07i)9-s + (0.572 − 0.208i)10-s + (3.42 + 5.93i)11-s + (6.00 − 10.3i)12-s + (−4.60 + 26.1i)13-s + (30.2 − 25.4i)14-s + (0.700 + 0.587i)15-s + (2.77 + 15.7i)16-s + (58.6 + 21.3i)17-s + ⋯ |
| L(s) = 1 | + (−0.664 − 0.241i)2-s + (−0.100 − 0.568i)3-s + (0.383 + 0.321i)4-s + (−0.0208 + 0.0175i)5-s + (−0.0708 + 0.402i)6-s + (−0.533 + 0.924i)7-s + (−0.176 − 0.306i)8-s + (−0.313 + 0.114i)9-s + (0.0181 − 0.00659i)10-s + (0.0939 + 0.162i)11-s + (0.144 − 0.250i)12-s + (−0.0983 + 0.557i)13-s + (0.577 − 0.484i)14-s + (0.0120 + 0.0101i)15-s + (0.0434 + 0.246i)16-s + (0.836 + 0.304i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.483 - 0.875i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.483 - 0.875i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.700193 + 0.412950i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.700193 + 0.412950i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.87 + 0.684i)T \) |
| 3 | \( 1 + (0.520 + 2.95i)T \) |
| 19 | \( 1 + (-4.75 - 82.6i)T \) |
| good | 5 | \( 1 + (0.233 - 0.195i)T + (21.7 - 123. i)T^{2} \) |
| 7 | \( 1 + (9.88 - 17.1i)T + (-171.5 - 297. i)T^{2} \) |
| 11 | \( 1 + (-3.42 - 5.93i)T + (-665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 + (4.60 - 26.1i)T + (-2.06e3 - 751. i)T^{2} \) |
| 17 | \( 1 + (-58.6 - 21.3i)T + (3.76e3 + 3.15e3i)T^{2} \) |
| 23 | \( 1 + (-83.0 - 69.6i)T + (2.11e3 + 1.19e4i)T^{2} \) |
| 29 | \( 1 + (-54.2 + 19.7i)T + (1.86e4 - 1.56e4i)T^{2} \) |
| 31 | \( 1 + (32.8 - 56.9i)T + (-1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 + 52.3T + 5.06e4T^{2} \) |
| 41 | \( 1 + (46.4 + 263. i)T + (-6.47e4 + 2.35e4i)T^{2} \) |
| 43 | \( 1 + (145. - 122. i)T + (1.38e4 - 7.82e4i)T^{2} \) |
| 47 | \( 1 + (314. - 114. i)T + (7.95e4 - 6.67e4i)T^{2} \) |
| 53 | \( 1 + (435. + 365. i)T + (2.58e4 + 1.46e5i)T^{2} \) |
| 59 | \( 1 + (-497. - 181. i)T + (1.57e5 + 1.32e5i)T^{2} \) |
| 61 | \( 1 + (361. + 303. i)T + (3.94e4 + 2.23e5i)T^{2} \) |
| 67 | \( 1 + (171. - 62.4i)T + (2.30e5 - 1.93e5i)T^{2} \) |
| 71 | \( 1 + (16.4 - 13.8i)T + (6.21e4 - 3.52e5i)T^{2} \) |
| 73 | \( 1 + (-31.7 - 179. i)T + (-3.65e5 + 1.33e5i)T^{2} \) |
| 79 | \( 1 + (-41.3 - 234. i)T + (-4.63e5 + 1.68e5i)T^{2} \) |
| 83 | \( 1 + (536. - 929. i)T + (-2.85e5 - 4.95e5i)T^{2} \) |
| 89 | \( 1 + (71.0 - 402. i)T + (-6.62e5 - 2.41e5i)T^{2} \) |
| 97 | \( 1 + (-1.52e3 - 556. i)T + (6.99e5 + 5.86e5i)T^{2} \) |
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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
−12.93122217277829315567111770327, −12.19154619862166318304520168089, −11.33319758903378012714473535812, −9.954613320504170026960568551993, −9.047182404191114962673170455902, −7.915303274390775082614122230147, −6.74226944887199350482840302322, −5.54113044981806707279916246630, −3.28572916505434985591243087300, −1.65901739064478055856622103046,
0.56361793554904031377263711567, 3.14463127626945602005109966588, 4.80557901553784934562426628432, 6.33007036773161434598231607071, 7.46292314084484282859255352635, 8.706345176497968954261428875798, 9.873773842901368343947394957623, 10.50762008737154484249611231096, 11.61048853882922863144343789877, 12.97170534373342356333852978341
