L(s) = 1 | + (−1.93 + 4.82i)3-s + (3.83 − 6.63i)5-s + (−19.4 − 18.6i)9-s + (−1.13 + 0.657i)11-s − 23.9i·13-s + (24.5 + 31.3i)15-s + (29.5 + 51.1i)17-s + (−24.4 − 14.1i)19-s + (−65.7 − 37.9i)23-s + (33.1 + 57.3i)25-s + (127. − 57.6i)27-s + 302. i·29-s + (−80.9 + 46.7i)31-s + (−0.960 − 6.76i)33-s + (−133. + 231. i)37-s + ⋯ |
L(s) = 1 | + (−0.373 + 0.927i)3-s + (0.342 − 0.593i)5-s + (−0.721 − 0.692i)9-s + (−0.0312 + 0.0180i)11-s − 0.511i·13-s + (0.422 + 0.539i)15-s + (0.421 + 0.730i)17-s + (−0.295 − 0.170i)19-s + (−0.595 − 0.344i)23-s + (0.264 + 0.458i)25-s + (0.911 − 0.410i)27-s + 1.93i·29-s + (−0.469 + 0.270i)31-s + (−0.00506 − 0.0356i)33-s + (−0.592 + 1.02i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.716 - 0.697i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.716 - 0.697i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.9942253237\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9942253237\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.93 - 4.82i)T \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (-3.83 + 6.63i)T + (-62.5 - 108. i)T^{2} \) |
| 11 | \( 1 + (1.13 - 0.657i)T + (665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 + 23.9iT - 2.19e3T^{2} \) |
| 17 | \( 1 + (-29.5 - 51.1i)T + (-2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (24.4 + 14.1i)T + (3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + (65.7 + 37.9i)T + (6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 - 302. iT - 2.43e4T^{2} \) |
| 31 | \( 1 + (80.9 - 46.7i)T + (1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 + (133. - 231. i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 - 142.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 284.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (-104. + 181. i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + (545. - 314. i)T + (7.44e4 - 1.28e5i)T^{2} \) |
| 59 | \( 1 + (365. + 632. i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-471. - 272. i)T + (1.13e5 + 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-240. - 416. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 + 46.5iT - 3.57e5T^{2} \) |
| 73 | \( 1 + (834. - 481. i)T + (1.94e5 - 3.36e5i)T^{2} \) |
| 79 | \( 1 + (630. - 1.09e3i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 + 841.T + 5.71e5T^{2} \) |
| 89 | \( 1 + (641. - 1.11e3i)T + (-3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 - 60.2iT - 9.12e5T^{2} \) |
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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
−10.57292109879873921705633707616, −9.818390366122142780201410943540, −8.937783906561681495987105694289, −8.278203469699596840239559216974, −6.90387006212063989190654728875, −5.76601934949868446676857850990, −5.15197484237339126709449512883, −4.11386457500128775679026703295, −3.02529783983119549414170896596, −1.28976444465721412830915101497,
0.31702287366419169593337501914, 1.84939272370696029819046968394, 2.79668725983549707390875187100, 4.31983953306208645999932964764, 5.68748087370114297303748772121, 6.28391309877355702659285460812, 7.27401321978065520951884777165, 7.915953784919336091781978494304, 9.093225507380778113135898412367, 10.04482151052292472488931221204