| L(s) = 1 | + (1.87 + 0.684i)2-s + (0.520 + 2.95i)3-s + (3.06 + 2.57i)4-s + (−11.5 + 9.72i)5-s + (−1.04 + 5.90i)6-s + (−2.24 + 3.88i)7-s + (4.00 + 6.92i)8-s + (−8.45 + 3.07i)9-s + (−28.4 + 10.3i)10-s + (−24.0 − 41.6i)11-s + (−6.00 + 10.3i)12-s + (−13.5 + 76.9i)13-s + (−6.86 + 5.76i)14-s + (−34.7 − 29.1i)15-s + (2.77 + 15.7i)16-s + (31.2 + 11.3i)17-s + ⋯ |
| L(s) = 1 | + (0.664 + 0.241i)2-s + (0.100 + 0.568i)3-s + (0.383 + 0.321i)4-s + (−1.03 + 0.869i)5-s + (−0.0708 + 0.402i)6-s + (−0.120 + 0.209i)7-s + (0.176 + 0.306i)8-s + (−0.313 + 0.114i)9-s + (−0.898 + 0.327i)10-s + (−0.659 − 1.14i)11-s + (−0.144 + 0.250i)12-s + (−0.289 + 1.64i)13-s + (−0.131 + 0.109i)14-s + (−0.598 − 0.501i)15-s + (0.0434 + 0.246i)16-s + (0.445 + 0.162i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.737 - 0.674i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.737 - 0.674i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.602703 + 1.55207i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.602703 + 1.55207i\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 + (-81.5 - 14.6i)T \) |
| good | 5 | \( 1 + (11.5 - 9.72i)T + (21.7 - 123. i)T^{2} \) |
| 7 | \( 1 + (2.24 - 3.88i)T + (-171.5 - 297. i)T^{2} \) |
| 11 | \( 1 + (24.0 + 41.6i)T + (-665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 + (13.5 - 76.9i)T + (-2.06e3 - 751. i)T^{2} \) |
| 17 | \( 1 + (-31.2 - 11.3i)T + (3.76e3 + 3.15e3i)T^{2} \) |
| 23 | \( 1 + (-47.5 - 39.9i)T + (2.11e3 + 1.19e4i)T^{2} \) |
| 29 | \( 1 + (-100. + 36.5i)T + (1.86e4 - 1.56e4i)T^{2} \) |
| 31 | \( 1 + (149. - 258. i)T + (-1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 - 159.T + 5.06e4T^{2} \) |
| 41 | \( 1 + (13.5 + 76.8i)T + (-6.47e4 + 2.35e4i)T^{2} \) |
| 43 | \( 1 + (-356. + 299. i)T + (1.38e4 - 7.82e4i)T^{2} \) |
| 47 | \( 1 + (-231. + 84.4i)T + (7.95e4 - 6.67e4i)T^{2} \) |
| 53 | \( 1 + (223. + 187. i)T + (2.58e4 + 1.46e5i)T^{2} \) |
| 59 | \( 1 + (376. + 137. i)T + (1.57e5 + 1.32e5i)T^{2} \) |
| 61 | \( 1 + (369. + 310. i)T + (3.94e4 + 2.23e5i)T^{2} \) |
| 67 | \( 1 + (-832. + 303. i)T + (2.30e5 - 1.93e5i)T^{2} \) |
| 71 | \( 1 + (119. - 100. i)T + (6.21e4 - 3.52e5i)T^{2} \) |
| 73 | \( 1 + (-148. - 841. i)T + (-3.65e5 + 1.33e5i)T^{2} \) |
| 79 | \( 1 + (-40.1 - 227. i)T + (-4.63e5 + 1.68e5i)T^{2} \) |
| 83 | \( 1 + (292. - 506. i)T + (-2.85e5 - 4.95e5i)T^{2} \) |
| 89 | \( 1 + (-129. + 732. i)T + (-6.62e5 - 2.41e5i)T^{2} \) |
| 97 | \( 1 + (1.23e3 + 448. i)T + (6.99e5 + 5.86e5i)T^{2} \) |
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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
−13.96243642757879924528175220769, −12.32403024116476594550302087536, −11.41694003177090711052706966898, −10.72028680589904119343782421459, −9.171196114161429883050096450094, −7.86818964210866011863976842407, −6.82843097319399667488827072087, −5.40624502151014776166177290694, −3.94148933351954698515517429297, −2.96903548331932798473731443720,
0.73122201504786923065587122082, 2.88334023744672905854417940732, 4.46469661862188195602662469881, 5.56037763225221998701318671222, 7.42045492679128993660032093545, 7.923785999720496745233560316039, 9.589871703114832567217733155604, 10.86488881786454915501013786575, 12.13376010200419027016867637450, 12.62375326017931494780717198111
