| L(s) = 1 | + (3.23 + 2.35i)2-s + 4.64i·3-s + (4.94 + 15.2i)4-s + (−24.0 − 74.1i)5-s + (−10.9 + 15.0i)6-s + (17.3 + 23.9i)7-s + (−19.7 + 60.8i)8-s + 221.·9-s + (96.3 − 296. i)10-s + (580. + 188. i)11-s + (−70.6 + 22.9i)12-s + (443. − 610. i)13-s + 118. i·14-s + (344. − 111. i)15-s + (−207. + 150. i)16-s + (−926. − 301. i)17-s + ⋯ |
| L(s) = 1 | + (0.572 + 0.415i)2-s + 0.297i·3-s + (0.154 + 0.475i)4-s + (−0.430 − 1.32i)5-s + (−0.123 + 0.170i)6-s + (0.134 + 0.184i)7-s + (−0.109 + 0.336i)8-s + 0.911·9-s + (0.304 − 0.937i)10-s + (1.44 + 0.469i)11-s + (−0.141 + 0.0460i)12-s + (0.728 − 1.00i)13-s + 0.161i·14-s + (0.394 − 0.128i)15-s + (−0.202 + 0.146i)16-s + (−0.777 − 0.252i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 82 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.967 - 0.254i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 82 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.967 - 0.254i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.63309 + 0.340303i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.63309 + 0.340303i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-3.23 - 2.35i)T \) |
| 41 | \( 1 + (1.05e4 - 2.03e3i)T \) |
| good | 3 | \( 1 - 4.64iT - 243T^{2} \) |
| 5 | \( 1 + (24.0 + 74.1i)T + (-2.52e3 + 1.83e3i)T^{2} \) |
| 7 | \( 1 + (-17.3 - 23.9i)T + (-5.19e3 + 1.59e4i)T^{2} \) |
| 11 | \( 1 + (-580. - 188. i)T + (1.30e5 + 9.46e4i)T^{2} \) |
| 13 | \( 1 + (-443. + 610. i)T + (-1.14e5 - 3.53e5i)T^{2} \) |
| 17 | \( 1 + (926. + 301. i)T + (1.14e6 + 8.34e5i)T^{2} \) |
| 19 | \( 1 + (-1.55e3 - 2.13e3i)T + (-7.65e5 + 2.35e6i)T^{2} \) |
| 23 | \( 1 + (774. + 562. i)T + (1.98e6 + 6.12e6i)T^{2} \) |
| 29 | \( 1 + (-4.94e3 + 1.60e3i)T + (1.65e7 - 1.20e7i)T^{2} \) |
| 31 | \( 1 + (-2.30e3 + 7.08e3i)T + (-2.31e7 - 1.68e7i)T^{2} \) |
| 37 | \( 1 + (2.34e3 + 7.20e3i)T + (-5.61e7 + 4.07e7i)T^{2} \) |
| 43 | \( 1 + (-9.66e3 - 7.02e3i)T + (4.54e7 + 1.39e8i)T^{2} \) |
| 47 | \( 1 + (1.64e4 - 2.26e4i)T + (-7.08e7 - 2.18e8i)T^{2} \) |
| 53 | \( 1 + (-3.02e3 + 983. i)T + (3.38e8 - 2.45e8i)T^{2} \) |
| 59 | \( 1 + (5.41e3 + 3.93e3i)T + (2.20e8 + 6.79e8i)T^{2} \) |
| 61 | \( 1 + (2.68e3 - 1.95e3i)T + (2.60e8 - 8.03e8i)T^{2} \) |
| 67 | \( 1 + (4.04e4 - 1.31e4i)T + (1.09e9 - 7.93e8i)T^{2} \) |
| 71 | \( 1 + (-2.80e4 - 9.10e3i)T + (1.45e9 + 1.06e9i)T^{2} \) |
| 73 | \( 1 + 6.43e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 1.49e4iT - 3.07e9T^{2} \) |
| 83 | \( 1 + 1.00e5T + 3.93e9T^{2} \) |
| 89 | \( 1 + (-1.45e4 - 2.00e4i)T + (-1.72e9 + 5.31e9i)T^{2} \) |
| 97 | \( 1 + (-1.43e5 + 4.67e4i)T + (6.94e9 - 5.04e9i)T^{2} \) |
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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
−13.28781811975456263896586186135, −12.42369941915093907883872917769, −11.64327656052927023940595412716, −9.902545435341323280126539295231, −8.753752316783346571737477775138, −7.68310251515795602503803960815, −6.11751890419254594857707320424, −4.70288431875959583310723610946, −3.86699571458638169179690514783, −1.23742724878627809136843232651,
1.42320588137139546300682838197, 3.24556773552467746803743847860, 4.38193275104667474407997485122, 6.62346295835287948153518336341, 6.92544560526384512434230566846, 8.904152620087790640530170829496, 10.34229717172401691964040031842, 11.35075185484473045835283664834, 11.96565573057332561948067951752, 13.61843192032060161985538258014
