| L(s) = 1 | + (15.0 + 5.47i)2-s + (−19.9 − 113. i)3-s + (196. + 164. i)4-s + (−800. + 671. i)5-s + (319. − 1.81e3i)6-s + (928. − 1.60e3i)7-s + (2.04e3 + 3.54e3i)8-s + (6.04e3 − 2.19e3i)9-s + (−1.57e4 + 5.71e3i)10-s + (−2.41e4 − 4.18e4i)11-s + (1.47e4 − 2.55e4i)12-s + (1.85e4 − 1.05e5i)13-s + (2.27e4 − 1.91e4i)14-s + (9.21e4 + 7.72e4i)15-s + (1.13e4 + 6.45e4i)16-s + (−5.68e5 − 2.06e5i)17-s + ⋯ |
| L(s) = 1 | + (0.664 + 0.241i)2-s + (−0.142 − 0.808i)3-s + (0.383 + 0.321i)4-s + (−0.572 + 0.480i)5-s + (0.100 − 0.571i)6-s + (0.146 − 0.253i)7-s + (0.176 + 0.306i)8-s + (0.306 − 0.111i)9-s + (−0.496 + 0.180i)10-s + (−0.498 − 0.862i)11-s + (0.205 − 0.355i)12-s + (0.180 − 1.02i)13-s + (0.158 − 0.132i)14-s + (0.469 + 0.394i)15-s + (0.0434 + 0.246i)16-s + (−1.65 − 0.601i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.490 + 0.871i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.490 + 0.871i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.803378 - 1.37370i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.803378 - 1.37370i\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-15.0 - 5.47i)T \) |
| 19 | \( 1 + (5.63e5 + 7.04e4i)T \) |
| good | 3 | \( 1 + (19.9 + 113. i)T + (-1.84e4 + 6.73e3i)T^{2} \) |
| 5 | \( 1 + (800. - 671. i)T + (3.39e5 - 1.92e6i)T^{2} \) |
| 7 | \( 1 + (-928. + 1.60e3i)T + (-2.01e7 - 3.49e7i)T^{2} \) |
| 11 | \( 1 + (2.41e4 + 4.18e4i)T + (-1.17e9 + 2.04e9i)T^{2} \) |
| 13 | \( 1 + (-1.85e4 + 1.05e5i)T + (-9.96e9 - 3.62e9i)T^{2} \) |
| 17 | \( 1 + (5.68e5 + 2.06e5i)T + (9.08e10 + 7.62e10i)T^{2} \) |
| 23 | \( 1 + (6.06e5 + 5.09e5i)T + (3.12e11 + 1.77e12i)T^{2} \) |
| 29 | \( 1 + (-3.58e6 + 1.30e6i)T + (1.11e13 - 9.32e12i)T^{2} \) |
| 31 | \( 1 + (-2.15e6 + 3.73e6i)T + (-1.32e13 - 2.28e13i)T^{2} \) |
| 37 | \( 1 - 1.33e7T + 1.29e14T^{2} \) |
| 41 | \( 1 + (5.01e6 + 2.84e7i)T + (-3.07e14 + 1.11e14i)T^{2} \) |
| 43 | \( 1 + (1.92e7 - 1.61e7i)T + (8.72e13 - 4.94e14i)T^{2} \) |
| 47 | \( 1 + (-1.01e7 + 3.68e6i)T + (8.57e14 - 7.19e14i)T^{2} \) |
| 53 | \( 1 + (-2.44e7 - 2.05e7i)T + (5.72e14 + 3.24e15i)T^{2} \) |
| 59 | \( 1 + (7.88e6 + 2.87e6i)T + (6.63e15 + 5.56e15i)T^{2} \) |
| 61 | \( 1 + (-1.05e8 - 8.87e7i)T + (2.03e15 + 1.15e16i)T^{2} \) |
| 67 | \( 1 + (9.83e7 - 3.57e7i)T + (2.08e16 - 1.74e16i)T^{2} \) |
| 71 | \( 1 + (3.60e7 - 3.02e7i)T + (7.96e15 - 4.51e16i)T^{2} \) |
| 73 | \( 1 + (-2.03e7 - 1.15e8i)T + (-5.53e16 + 2.01e16i)T^{2} \) |
| 79 | \( 1 + (5.34e7 + 3.02e8i)T + (-1.12e17 + 4.09e16i)T^{2} \) |
| 83 | \( 1 + (8.66e7 - 1.50e8i)T + (-9.34e16 - 1.61e17i)T^{2} \) |
| 89 | \( 1 + (-1.29e8 + 7.36e8i)T + (-3.29e17 - 1.19e17i)T^{2} \) |
| 97 | \( 1 + (9.37e8 + 3.41e8i)T + (5.82e17 + 4.88e17i)T^{2} \) |
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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
−13.62364476346823406558421286372, −12.96651334257797937464262409185, −11.63370315037433885469356812665, −10.61897775184302631480718431552, −8.313265975641408885058309479815, −7.20601079978091283328485172695, −6.09720151891347711516814437865, −4.24776389083473277731443257260, −2.57865949719211376849064923244, −0.45332920725066502054455598108,
1.99452416215335016881001341456, 4.17494000493623072213091584360, 4.73527501728422782433110861865, 6.65711375830855578046974866677, 8.492672157415380045682202654204, 9.985363028562379788502813121820, 11.12599467050637962315891691282, 12.29706585262260854755479146984, 13.36714235052355160846726309912, 14.94999890304595749911037485658
