| L(s) = 1 | + (−3.23 − 2.35i)2-s + (−2.85 − 27.1i)3-s + (4.94 + 15.2i)4-s + (38.2 + 66.2i)5-s + (−54.6 + 94.6i)6-s + (−128. − 27.2i)7-s + (19.7 − 60.8i)8-s + (−492. + 104. i)9-s + (31.9 − 304. i)10-s + (−495. + 550. i)11-s + (399. − 177. i)12-s + (760. + 338. i)13-s + (350. + 389. i)14-s + (1.69e3 − 1.22e3i)15-s + (−207. + 150. i)16-s + (83.9 + 93.2i)17-s + ⋯ |
| L(s) = 1 | + (−0.572 − 0.415i)2-s + (−0.183 − 1.74i)3-s + (0.154 + 0.475i)4-s + (0.683 + 1.18i)5-s + (−0.619 + 1.07i)6-s + (−0.989 − 0.210i)7-s + (0.109 − 0.336i)8-s + (−2.02 + 0.431i)9-s + (0.101 − 0.961i)10-s + (−1.23 + 1.37i)11-s + (0.800 − 0.356i)12-s + (1.24 + 0.555i)13-s + (0.478 + 0.531i)14-s + (1.93 − 1.40i)15-s + (−0.202 + 0.146i)16-s + (0.0704 + 0.0782i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 62 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.522 - 0.852i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 62 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.522 - 0.852i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.449361 + 0.251554i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.449361 + 0.251554i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (3.23 + 2.35i)T \) |
| 31 | \( 1 + (-696. + 5.30e3i)T \) |
| good | 3 | \( 1 + (2.85 + 27.1i)T + (-237. + 50.5i)T^{2} \) |
| 5 | \( 1 + (-38.2 - 66.2i)T + (-1.56e3 + 2.70e3i)T^{2} \) |
| 7 | \( 1 + (128. + 27.2i)T + (1.53e4 + 6.83e3i)T^{2} \) |
| 11 | \( 1 + (495. - 550. i)T + (-1.68e4 - 1.60e5i)T^{2} \) |
| 13 | \( 1 + (-760. - 338. i)T + (2.48e5 + 2.75e5i)T^{2} \) |
| 17 | \( 1 + (-83.9 - 93.2i)T + (-1.48e5 + 1.41e6i)T^{2} \) |
| 19 | \( 1 + (176. - 78.7i)T + (1.65e6 - 1.84e6i)T^{2} \) |
| 23 | \( 1 + (110. - 341. i)T + (-5.20e6 - 3.78e6i)T^{2} \) |
| 29 | \( 1 + (2.39e3 + 1.74e3i)T + (6.33e6 + 1.95e7i)T^{2} \) |
| 37 | \( 1 + (7.72e3 - 1.33e4i)T + (-3.46e7 - 6.00e7i)T^{2} \) |
| 41 | \( 1 + (1.65e3 - 1.57e4i)T + (-1.13e8 - 2.40e7i)T^{2} \) |
| 43 | \( 1 + (5.33e3 - 2.37e3i)T + (9.83e7 - 1.09e8i)T^{2} \) |
| 47 | \( 1 + (-3.04e3 + 2.21e3i)T + (7.08e7 - 2.18e8i)T^{2} \) |
| 53 | \( 1 + (901. - 191. i)T + (3.82e8 - 1.70e8i)T^{2} \) |
| 59 | \( 1 + (-4.48e3 - 4.27e4i)T + (-6.99e8 + 1.48e8i)T^{2} \) |
| 61 | \( 1 + 2.27e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + (3.42e4 + 5.94e4i)T + (-6.75e8 + 1.16e9i)T^{2} \) |
| 71 | \( 1 + (-1.64e4 + 3.49e3i)T + (1.64e9 - 7.33e8i)T^{2} \) |
| 73 | \( 1 + (-1.18e4 + 1.31e4i)T + (-2.16e8 - 2.06e9i)T^{2} \) |
| 79 | \( 1 + (651. + 723. i)T + (-3.21e8 + 3.06e9i)T^{2} \) |
| 83 | \( 1 + (8.44e3 - 8.03e4i)T + (-3.85e9 - 8.18e8i)T^{2} \) |
| 89 | \( 1 + (2.41e3 + 7.42e3i)T + (-4.51e9 + 3.28e9i)T^{2} \) |
| 97 | \( 1 + (-1.99e4 - 6.13e4i)T + (-6.94e9 + 5.04e9i)T^{2} \) |
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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
−13.53194478646132328218563228888, −13.21969676036057415826619572801, −12.00780284123360263913066105673, −10.77167701783957145148598406211, −9.737732359850233975987276079780, −7.956171049499192126651489913590, −6.90611312577968354472728132508, −6.21134872296549243433959526484, −2.87043302717084185626193879501, −1.74033323715192074174822682297,
0.28478923526730443424735733621, 3.39118948183667089626816904294, 5.30093567753212811668615980251, 5.84503665864239215832245109970, 8.577119720499981628485021893733, 9.045557491070914847252117299957, 10.23392188931306222770083009580, 10.92581078602051360453698530462, 12.85509381087283890954060536949, 13.95208002133760032245660214121
