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
−10.11761105454689276631361952214, −9.168337791262111515838128531136, −8.134519810095323496481442998309, −7.62704026023687478814582723183, −6.91230765352779981113242844653, −5.65285825341140638875953896837, −4.65059682982139770125851951021, −3.59609981640647122666358399276, −2.55519829918482498309272213770, −1.18665425438903847191621187816,
1.42452957051676808160622173652, 2.55877759232529329467915660013, 3.84827666655221956893589492222, 5.04083877349462980400541246632, 5.34002982421341085511330747644, 6.91238773105931880189142470622, 7.84970760359577721088621718879, 8.489026825822248793019899499994, 9.144516544889139529899777924570, 9.983680287689499624931312212745