| L(s) = 1 | + (−0.909 − 1.08i)2-s + (−1.73 + 4.75i)3-s + (−0.347 + 1.96i)4-s + (0.678 + 3.85i)5-s + (6.72 − 2.44i)6-s + (1.77 − 3.07i)7-s + (2.44 − 1.41i)8-s + (−12.7 − 10.6i)9-s + (3.55 − 4.23i)10-s + (9.47 + 16.4i)11-s + (−8.76 − 5.05i)12-s + (−5.36 − 14.7i)13-s + (−4.94 + 0.872i)14-s + (−19.4 − 3.43i)15-s + (−3.75 − 1.36i)16-s + (15.2 − 12.8i)17-s + ⋯ |
| L(s) = 1 | + (−0.454 − 0.541i)2-s + (−0.576 + 1.58i)3-s + (−0.0868 + 0.492i)4-s + (0.135 + 0.770i)5-s + (1.12 − 0.407i)6-s + (0.253 − 0.439i)7-s + (0.306 − 0.176i)8-s + (−1.41 − 1.18i)9-s + (0.355 − 0.423i)10-s + (0.861 + 1.49i)11-s + (−0.730 − 0.421i)12-s + (−0.412 − 1.13i)13-s + (−0.353 + 0.0622i)14-s + (−1.29 − 0.228i)15-s + (−0.234 − 0.0855i)16-s + (0.897 − 0.753i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.290 - 0.956i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.290 - 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.595182 + 0.441121i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.595182 + 0.441121i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.909 + 1.08i)T \) |
| 19 | \( 1 + (-18.1 - 5.51i)T \) |
| good | 3 | \( 1 + (1.73 - 4.75i)T + (-6.89 - 5.78i)T^{2} \) |
| 5 | \( 1 + (-0.678 - 3.85i)T + (-23.4 + 8.55i)T^{2} \) |
| 7 | \( 1 + (-1.77 + 3.07i)T + (-24.5 - 42.4i)T^{2} \) |
| 11 | \( 1 + (-9.47 - 16.4i)T + (-60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + (5.36 + 14.7i)T + (-129. + 108. i)T^{2} \) |
| 17 | \( 1 + (-15.2 + 12.8i)T + (50.1 - 284. i)T^{2} \) |
| 23 | \( 1 + (1.88 - 10.6i)T + (-497. - 180. i)T^{2} \) |
| 29 | \( 1 + (14.3 - 17.1i)T + (-146. - 828. i)T^{2} \) |
| 31 | \( 1 + (18.6 + 10.7i)T + (480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + 19.0iT - 1.36e3T^{2} \) |
| 41 | \( 1 + (3.04 - 8.36i)T + (-1.28e3 - 1.08e3i)T^{2} \) |
| 43 | \( 1 + (4.24 + 24.0i)T + (-1.73e3 + 632. i)T^{2} \) |
| 47 | \( 1 + (59.2 + 49.6i)T + (383. + 2.17e3i)T^{2} \) |
| 53 | \( 1 + (-28.8 - 5.08i)T + (2.63e3 + 960. i)T^{2} \) |
| 59 | \( 1 + (30.8 + 36.8i)T + (-604. + 3.42e3i)T^{2} \) |
| 61 | \( 1 + (-12.7 + 72.3i)T + (-3.49e3 - 1.27e3i)T^{2} \) |
| 67 | \( 1 + (49.1 - 58.5i)T + (-779. - 4.42e3i)T^{2} \) |
| 71 | \( 1 + (-108. + 19.0i)T + (4.73e3 - 1.72e3i)T^{2} \) |
| 73 | \( 1 + (-34.6 - 12.6i)T + (4.08e3 + 3.42e3i)T^{2} \) |
| 79 | \( 1 + (-10.6 + 29.2i)T + (-4.78e3 - 4.01e3i)T^{2} \) |
| 83 | \( 1 + (-4.45 + 7.72i)T + (-3.44e3 - 5.96e3i)T^{2} \) |
| 89 | \( 1 + (6.54 + 17.9i)T + (-6.06e3 + 5.09e3i)T^{2} \) |
| 97 | \( 1 + (39.7 + 47.3i)T + (-1.63e3 + 9.26e3i)T^{2} \) |
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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
−16.49960098336129485609971194939, −15.20081472387965882385818301438, −14.37415071398102276137050536982, −12.24332465305664831366487411116, −11.12062848794446219290979934686, −10.09024919776350361282861090948, −9.551798562803918288882360391879, −7.34489270248996031107249241313, −5.12775225270210316207136833637, −3.52521927355373123101386514732,
1.26769890497188247362476622076, 5.52624643376194692628622155648, 6.60590714169901426520733115921, 8.036332627789628972626508737032, 9.104631090714001277205163665455, 11.35477470629257280277199877110, 12.20573181262354328883283790330, 13.49490625449998330052830264257, 14.45368775345815833081772960272, 16.56922644764578966290234925922
