L(s) = 1 | + (0.939 + 0.342i)3-s + (−0.233 − 1.32i)5-s + (3.20 + 1.85i)7-s + (0.766 + 0.642i)9-s + (−1.31 + 0.761i)11-s + (0.286 + 0.788i)13-s + (0.233 − 1.32i)15-s + (0.124 − 0.104i)17-s + (4.35 + 0.0632i)19-s + (2.37 + 2.83i)21-s + (1.85 + 0.327i)23-s + (2.99 − 1.08i)25-s + (0.500 + 0.866i)27-s + (−6.75 + 8.04i)29-s + (2.95 − 5.11i)31-s + ⋯ |
L(s) = 1 | + (0.542 + 0.197i)3-s + (−0.104 − 0.593i)5-s + (1.21 + 0.699i)7-s + (0.255 + 0.214i)9-s + (−0.397 + 0.229i)11-s + (0.0795 + 0.218i)13-s + (0.0604 − 0.342i)15-s + (0.0301 − 0.0253i)17-s + (0.999 + 0.0144i)19-s + (0.519 + 0.618i)21-s + (0.387 + 0.0683i)23-s + (0.598 − 0.217i)25-s + (0.0962 + 0.166i)27-s + (−1.25 + 1.49i)29-s + (0.529 − 0.917i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.944 - 0.327i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.944 - 0.327i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.11698 + 0.356765i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.11698 + 0.356765i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.939 - 0.342i)T \) |
| 19 | \( 1 + (-4.35 - 0.0632i)T \) |
good | 5 | \( 1 + (0.233 + 1.32i)T + (-4.69 + 1.71i)T^{2} \) |
| 7 | \( 1 + (-3.20 - 1.85i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (1.31 - 0.761i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.286 - 0.788i)T + (-9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (-0.124 + 0.104i)T + (2.95 - 16.7i)T^{2} \) |
| 23 | \( 1 + (-1.85 - 0.327i)T + (21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (6.75 - 8.04i)T + (-5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (-2.95 + 5.11i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 3.67iT - 37T^{2} \) |
| 41 | \( 1 + (-0.788 + 2.16i)T + (-31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + (2.89 - 0.509i)T + (40.4 - 14.7i)T^{2} \) |
| 47 | \( 1 + (-5.63 + 6.71i)T + (-8.16 - 46.2i)T^{2} \) |
| 53 | \( 1 + (6.69 + 1.18i)T + (49.8 + 18.1i)T^{2} \) |
| 59 | \( 1 + (-9.06 + 7.60i)T + (10.2 - 58.1i)T^{2} \) |
| 61 | \( 1 + (-0.741 + 4.20i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (-8.57 - 7.19i)T + (11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (-1.29 - 7.34i)T + (-66.7 + 24.2i)T^{2} \) |
| 73 | \( 1 + (-6.53 - 2.37i)T + (55.9 + 46.9i)T^{2} \) |
| 79 | \( 1 + (12.9 + 4.72i)T + (60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (-0.134 - 0.0775i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-1.92 - 5.30i)T + (-68.1 + 57.2i)T^{2} \) |
| 97 | \( 1 + (5.19 + 6.19i)T + (-16.8 + 95.5i)T^{2} \) |
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
−9.983680287689499624931312212745, −9.144516544889139529899777924570, −8.489026825822248793019899499994, −7.84970760359577721088621718879, −6.91238773105931880189142470622, −5.34002982421341085511330747644, −5.04083877349462980400541246632, −3.84827666655221956893589492222, −2.55877759232529329467915660013, −1.42452957051676808160622173652,
1.18665425438903847191621187816, 2.55519829918482498309272213770, 3.59609981640647122666358399276, 4.65059682982139770125851951021, 5.65285825341140638875953896837, 6.91230765352779981113242844653, 7.62704026023687478814582723183, 8.134519810095323496481442998309, 9.168337791262111515838128531136, 10.11761105454689276631361952214