| L(s) = 1 | + (−1.61 + 0.618i)3-s − i·5-s + 2.07i·7-s + (2.23 − 2.00i)9-s − 11-s + 6.58·13-s + (0.618 + 1.61i)15-s − 4.07i·17-s − 4.18i·19-s + (−1.28 − 3.35i)21-s − 6.70·23-s − 25-s + (−2.38 + 4.61i)27-s + 2.18i·29-s − 8.70i·31-s + ⋯ |
| L(s) = 1 | + (−0.934 + 0.356i)3-s − 0.447i·5-s + 0.783i·7-s + (0.745 − 0.666i)9-s − 0.301·11-s + 1.82·13-s + (0.159 + 0.417i)15-s − 0.987i·17-s − 0.960i·19-s + (−0.279 − 0.731i)21-s − 1.39·23-s − 0.200·25-s + (−0.458 + 0.888i)27-s + 0.406i·29-s − 1.56i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.356 + 0.934i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.356 + 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.020556715\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.020556715\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.61 - 0.618i)T \) |
| 5 | \( 1 + iT \) |
| 11 | \( 1 + T \) |
| good | 7 | \( 1 - 2.07iT - 7T^{2} \) |
| 13 | \( 1 - 6.58T + 13T^{2} \) |
| 17 | \( 1 + 4.07iT - 17T^{2} \) |
| 19 | \( 1 + 4.18iT - 19T^{2} \) |
| 23 | \( 1 + 6.70T + 23T^{2} \) |
| 29 | \( 1 - 2.18iT - 29T^{2} \) |
| 31 | \( 1 + 8.70iT - 31T^{2} \) |
| 37 | \( 1 + 9.61T + 37T^{2} \) |
| 41 | \( 1 - 6.18iT - 41T^{2} \) |
| 43 | \( 1 - 11.2iT - 43T^{2} \) |
| 47 | \( 1 - 9.94T + 47T^{2} \) |
| 53 | \( 1 + 5.94iT - 53T^{2} \) |
| 59 | \( 1 + 0.532T + 59T^{2} \) |
| 61 | \( 1 - 0.704T + 61T^{2} \) |
| 67 | \( 1 + 4.14iT - 67T^{2} \) |
| 71 | \( 1 + 5.37T + 71T^{2} \) |
| 73 | \( 1 + 4.25T + 73T^{2} \) |
| 79 | \( 1 + 11.9iT - 79T^{2} \) |
| 83 | \( 1 - 2.25T + 83T^{2} \) |
| 89 | \( 1 + 17.5iT - 89T^{2} \) |
| 97 | \( 1 - 7.85T + 97T^{2} \) |
| 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
−8.863469826626804808370613273407, −8.055826399623313865592286561723, −7.06596119284546989625097657870, −6.06661933895449422408482006575, −5.77851487622195042405480764644, −4.83377905589336601537237264774, −4.11444843170446484315048976354, −3.05856772816147809838929956894, −1.71062457221034350879448793273, −0.43937626299655193506837503797,
1.10872049940344023890306750787, 2.01574884410321890921528670043, 3.79013236488572669552551565728, 3.92185223826073096678460767909, 5.40683158417529336610661200854, 5.94837867081613155903032166984, 6.64840994858648375580463531583, 7.35996597204259991010820254591, 8.158930766430738009907060779383, 8.823902177797321474893150281094
