| L(s) = 1 | + (0.694 + 3.93i)2-s + (19.8 − 16.6i)3-s + (−15.0 + 5.47i)4-s + (17.3 + 6.31i)5-s + (79.3 + 66.6i)6-s + (61.0 − 105. i)7-s + (−32 − 55.4i)8-s + (74.3 − 421. i)9-s + (−12.8 + 72.7i)10-s + (35.5 + 61.6i)11-s + (−207. + 358. i)12-s + (413. + 347. i)13-s + (458. + 167. i)14-s + (449. − 163. i)15-s + (196. − 164. i)16-s + (38.4 + 217. i)17-s + ⋯ |
| L(s) = 1 | + (0.122 + 0.696i)2-s + (1.27 − 1.06i)3-s + (−0.469 + 0.171i)4-s + (0.310 + 0.112i)5-s + (0.900 + 0.755i)6-s + (0.470 − 0.815i)7-s + (−0.176 − 0.306i)8-s + (0.305 − 1.73i)9-s + (−0.0405 + 0.229i)10-s + (0.0886 + 0.153i)11-s + (−0.415 + 0.719i)12-s + (0.679 + 0.569i)13-s + (0.625 + 0.227i)14-s + (0.515 − 0.187i)15-s + (0.191 − 0.160i)16-s + (0.0322 + 0.182i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 + 0.175i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.984 + 0.175i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.48977 - 0.220377i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.48977 - 0.220377i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.694 - 3.93i)T \) |
| 19 | \( 1 + (-511. - 1.48e3i)T \) |
| good | 3 | \( 1 + (-19.8 + 16.6i)T + (42.1 - 239. i)T^{2} \) |
| 5 | \( 1 + (-17.3 - 6.31i)T + (2.39e3 + 2.00e3i)T^{2} \) |
| 7 | \( 1 + (-61.0 + 105. i)T + (-8.40e3 - 1.45e4i)T^{2} \) |
| 11 | \( 1 + (-35.5 - 61.6i)T + (-8.05e4 + 1.39e5i)T^{2} \) |
| 13 | \( 1 + (-413. - 347. i)T + (6.44e4 + 3.65e5i)T^{2} \) |
| 17 | \( 1 + (-38.4 - 217. i)T + (-1.33e6 + 4.85e5i)T^{2} \) |
| 23 | \( 1 + (3.86e3 - 1.40e3i)T + (4.93e6 - 4.13e6i)T^{2} \) |
| 29 | \( 1 + (-964. + 5.46e3i)T + (-1.92e7 - 7.01e6i)T^{2} \) |
| 31 | \( 1 + (3.77e3 - 6.53e3i)T + (-1.43e7 - 2.47e7i)T^{2} \) |
| 37 | \( 1 - 6.11e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + (1.56e4 - 1.30e4i)T + (2.01e7 - 1.14e8i)T^{2} \) |
| 43 | \( 1 + (4.79e3 + 1.74e3i)T + (1.12e8 + 9.44e7i)T^{2} \) |
| 47 | \( 1 + (-912. + 5.17e3i)T + (-2.15e8 - 7.84e7i)T^{2} \) |
| 53 | \( 1 + (-3.23e3 + 1.17e3i)T + (3.20e8 - 2.68e8i)T^{2} \) |
| 59 | \( 1 + (-9.08e3 - 5.15e4i)T + (-6.71e8 + 2.44e8i)T^{2} \) |
| 61 | \( 1 + (-4.15e4 + 1.51e4i)T + (6.46e8 - 5.42e8i)T^{2} \) |
| 67 | \( 1 + (-1.13e4 + 6.40e4i)T + (-1.26e9 - 4.61e8i)T^{2} \) |
| 71 | \( 1 + (3.70e4 + 1.34e4i)T + (1.38e9 + 1.15e9i)T^{2} \) |
| 73 | \( 1 + (5.82e3 - 4.88e3i)T + (3.59e8 - 2.04e9i)T^{2} \) |
| 79 | \( 1 + (-3.19e4 + 2.67e4i)T + (5.34e8 - 3.03e9i)T^{2} \) |
| 83 | \( 1 + (-1.78e4 + 3.09e4i)T + (-1.96e9 - 3.41e9i)T^{2} \) |
| 89 | \( 1 + (-6.82e4 - 5.72e4i)T + (9.69e8 + 5.49e9i)T^{2} \) |
| 97 | \( 1 + (2.90e3 + 1.64e4i)T + (-8.06e9 + 2.93e9i)T^{2} \) |
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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
−14.85919514828050086203211985047, −13.89128208882876713057796083486, −13.54513909580916203283894510253, −12.01997587360054136885377178159, −9.898874393946851027383573688281, −8.392343140348219235029186223343, −7.58967571958328244899241686789, −6.30593812970964566921540048420, −3.83716146711718675635439896311, −1.65348924668444002075700189589,
2.28176336041672583499196088323, 3.73442439845071831422972529347, 5.32710765055701919181059014867, 8.255044744497218877786507522981, 9.129376552994174417192736271715, 10.17932464757210113758994821733, 11.50093322253077658605028732937, 13.16182125170411974017813634190, 14.18322356901204007592923354788, 15.13325275892449778723801451423
