| L(s) = 1 | + (−0.851 + 4.82i)2-s + (0.458 + 0.384i)3-s + (−15.0 − 5.49i)4-s + (6.94 − 2.52i)5-s + (−2.24 + 1.88i)6-s + (14.5 + 25.2i)7-s + (19.7 − 34.2i)8-s + (−4.62 − 26.2i)9-s + (6.29 + 35.7i)10-s + (26.3 − 45.7i)11-s + (−4.80 − 8.32i)12-s + (−30.9 + 26.0i)13-s + (−134. + 48.8i)14-s + (4.15 + 1.51i)15-s + (50.0 + 41.9i)16-s + (−0.0277 + 0.157i)17-s + ⋯ |
| L(s) = 1 | + (−0.301 + 1.70i)2-s + (0.0882 + 0.0740i)3-s + (−1.88 − 0.686i)4-s + (0.621 − 0.226i)5-s + (−0.153 + 0.128i)6-s + (0.786 + 1.36i)7-s + (0.872 − 1.51i)8-s + (−0.171 − 0.971i)9-s + (0.199 + 1.12i)10-s + (0.723 − 1.25i)11-s + (−0.115 − 0.200i)12-s + (−0.661 + 0.554i)13-s + (−2.56 + 0.933i)14-s + (0.0715 + 0.0260i)15-s + (0.781 + 0.655i)16-s + (−0.000395 + 0.00224i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.476 - 0.879i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.476 - 0.879i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.497272 + 0.834913i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.497272 + 0.834913i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 19 | \( 1 + (13.4 + 81.7i)T \) |
| good | 2 | \( 1 + (0.851 - 4.82i)T + (-7.51 - 2.73i)T^{2} \) |
| 3 | \( 1 + (-0.458 - 0.384i)T + (4.68 + 26.5i)T^{2} \) |
| 5 | \( 1 + (-6.94 + 2.52i)T + (95.7 - 80.3i)T^{2} \) |
| 7 | \( 1 + (-14.5 - 25.2i)T + (-171.5 + 297. i)T^{2} \) |
| 11 | \( 1 + (-26.3 + 45.7i)T + (-665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 + (30.9 - 26.0i)T + (381. - 2.16e3i)T^{2} \) |
| 17 | \( 1 + (0.0277 - 0.157i)T + (-4.61e3 - 1.68e3i)T^{2} \) |
| 23 | \( 1 + (39.4 + 14.3i)T + (9.32e3 + 7.82e3i)T^{2} \) |
| 29 | \( 1 + (-11.8 - 66.9i)T + (-2.29e4 + 8.34e3i)T^{2} \) |
| 31 | \( 1 + (-108. - 187. i)T + (-1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + 288.T + 5.06e4T^{2} \) |
| 41 | \( 1 + (24.4 + 20.4i)T + (1.19e4 + 6.78e4i)T^{2} \) |
| 43 | \( 1 + (-128. + 46.8i)T + (6.09e4 - 5.11e4i)T^{2} \) |
| 47 | \( 1 + (32.7 + 185. i)T + (-9.75e4 + 3.55e4i)T^{2} \) |
| 53 | \( 1 + (31.5 + 11.4i)T + (1.14e5 + 9.56e4i)T^{2} \) |
| 59 | \( 1 + (-50.5 + 286. i)T + (-1.92e5 - 7.02e4i)T^{2} \) |
| 61 | \( 1 + (10.9 + 4.00i)T + (1.73e5 + 1.45e5i)T^{2} \) |
| 67 | \( 1 + (-61.8 - 350. i)T + (-2.82e5 + 1.02e5i)T^{2} \) |
| 71 | \( 1 + (237. - 86.4i)T + (2.74e5 - 2.30e5i)T^{2} \) |
| 73 | \( 1 + (619. + 519. i)T + (6.75e4 + 3.83e5i)T^{2} \) |
| 79 | \( 1 + (-540. - 453. i)T + (8.56e4 + 4.85e5i)T^{2} \) |
| 83 | \( 1 + (-608. - 1.05e3i)T + (-2.85e5 + 4.95e5i)T^{2} \) |
| 89 | \( 1 + (-895. + 751. i)T + (1.22e5 - 6.94e5i)T^{2} \) |
| 97 | \( 1 + (69.0 - 391. i)T + (-8.57e5 - 3.12e5i)T^{2} \) |
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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
−17.90368293811860468012822633383, −17.21784062313970370627199098757, −15.84607431856518601505961073891, −14.78752822705444048120796019113, −13.93016927599919676452666539247, −11.87916840916491996052765983677, −9.199162985915140776562811729272, −8.622860819271479278195669629169, −6.51451448215949238147278493836, −5.30795312797621962920238450076,
1.89422877514160191127270379411, 4.38544348485857710567153099270, 7.78906971110019606873340229109, 9.884690990208406748667191346597, 10.56446462903933111489179809580, 11.99386035691662806676110181306, 13.43027296057373223494949327059, 14.37519980642387978300290591800, 17.15742414484962570428447036630, 17.70934285464968449903553812897
