| 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 | \( 1 + (39.7 - 47.3i)T + (-1.63e3 - 9.26e3i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−16.56922644764578966290234925922, −14.45368775345815833081772960272, −13.49490625449998330052830264257, −12.20573181262354328883283790330, −11.35477470629257280277199877110, −9.104631090714001277205163665455, −8.036332627789628972626508737032, −6.60590714169901426520733115921, −5.52624643376194692628622155648, −1.26769890497188247362476622076,
3.52521927355373123101386514732, 5.12775225270210316207136833637, 7.34489270248996031107249241313, 9.551798562803918288882360391879, 10.09024919776350361282861090948, 11.12062848794446219290979934686, 12.24332465305664831366487411116, 14.37415071398102276137050536982, 15.20081472387965882385818301438, 16.49960098336129485609971194939
