| 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
−15.13325275892449778723801451423, −14.18322356901204007592923354788, −13.16182125170411974017813634190, −11.50093322253077658605028732937, −10.17932464757210113758994821733, −9.129376552994174417192736271715, −8.255044744497218877786507522981, −5.32710765055701919181059014867, −3.73442439845071831422972529347, −2.28176336041672583499196088323,
1.65348924668444002075700189589, 3.83716146711718675635439896311, 6.30593812970964566921540048420, 7.58967571958328244899241686789, 8.392343140348219235029186223343, 9.898874393946851027383573688281, 12.01997587360054136885377178159, 13.54513909580916203283894510253, 13.89128208882876713057796083486, 14.85919514828050086203211985047
