| L(s) = 1 | + (0.415 + 0.909i)2-s + (0.959 − 0.281i)3-s + (−0.654 + 0.755i)4-s + (0.841 + 0.540i)5-s + (0.654 + 0.755i)6-s + (−0.668 − 4.64i)7-s + (−0.959 − 0.281i)8-s + (0.841 − 0.540i)9-s + (−0.142 + 0.989i)10-s + (0.596 − 1.30i)11-s + (−0.415 + 0.909i)12-s + (0.659 − 4.58i)13-s + (3.94 − 2.53i)14-s + (0.959 + 0.281i)15-s + (−0.142 − 0.989i)16-s + (2.16 + 2.49i)17-s + ⋯ |
| L(s) = 1 | + (0.293 + 0.643i)2-s + (0.553 − 0.162i)3-s + (−0.327 + 0.377i)4-s + (0.376 + 0.241i)5-s + (0.267 + 0.308i)6-s + (−0.252 − 1.75i)7-s + (−0.339 − 0.0996i)8-s + (0.280 − 0.180i)9-s + (−0.0450 + 0.313i)10-s + (0.179 − 0.393i)11-s + (−0.119 + 0.262i)12-s + (0.182 − 1.27i)13-s + (1.05 − 0.678i)14-s + (0.247 + 0.0727i)15-s + (−0.0355 − 0.247i)16-s + (0.525 + 0.606i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.982 + 0.185i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.982 + 0.185i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.09967 - 0.196944i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.09967 - 0.196944i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.415 - 0.909i)T \) |
| 3 | \( 1 + (-0.959 + 0.281i)T \) |
| 5 | \( 1 + (-0.841 - 0.540i)T \) |
| 23 | \( 1 + (2.66 - 3.98i)T \) |
| good | 7 | \( 1 + (0.668 + 4.64i)T + (-6.71 + 1.97i)T^{2} \) |
| 11 | \( 1 + (-0.596 + 1.30i)T + (-7.20 - 8.31i)T^{2} \) |
| 13 | \( 1 + (-0.659 + 4.58i)T + (-12.4 - 3.66i)T^{2} \) |
| 17 | \( 1 + (-2.16 - 2.49i)T + (-2.41 + 16.8i)T^{2} \) |
| 19 | \( 1 + (-2.75 + 3.18i)T + (-2.70 - 18.8i)T^{2} \) |
| 29 | \( 1 + (2.01 + 2.32i)T + (-4.12 + 28.7i)T^{2} \) |
| 31 | \( 1 + (-9.43 - 2.76i)T + (26.0 + 16.7i)T^{2} \) |
| 37 | \( 1 + (9.64 - 6.19i)T + (15.3 - 33.6i)T^{2} \) |
| 41 | \( 1 + (-2.56 - 1.65i)T + (17.0 + 37.2i)T^{2} \) |
| 43 | \( 1 + (4.11 - 1.20i)T + (36.1 - 23.2i)T^{2} \) |
| 47 | \( 1 - 0.295T + 47T^{2} \) |
| 53 | \( 1 + (-1.91 - 13.3i)T + (-50.8 + 14.9i)T^{2} \) |
| 59 | \( 1 + (-2.09 + 14.5i)T + (-56.6 - 16.6i)T^{2} \) |
| 61 | \( 1 + (-8.12 - 2.38i)T + (51.3 + 32.9i)T^{2} \) |
| 67 | \( 1 + (4.93 + 10.8i)T + (-43.8 + 50.6i)T^{2} \) |
| 71 | \( 1 + (-4.97 - 10.8i)T + (-46.4 + 53.6i)T^{2} \) |
| 73 | \( 1 + (-5.60 + 6.46i)T + (-10.3 - 72.2i)T^{2} \) |
| 79 | \( 1 + (1.95 - 13.5i)T + (-75.7 - 22.2i)T^{2} \) |
| 83 | \( 1 + (-7.86 + 5.05i)T + (34.4 - 75.4i)T^{2} \) |
| 89 | \( 1 + (5.33 - 1.56i)T + (74.8 - 48.1i)T^{2} \) |
| 97 | \( 1 + (8.67 + 5.57i)T + (40.2 + 88.2i)T^{2} \) |
| 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
−10.20067365874784280969768294759, −9.740341973562682041766462291230, −8.380502732780823880115970281158, −7.76597007398142189519511995992, −6.97379012710304110670813638323, −6.16074366252694133081212070641, −5.00293284396591741067649160299, −3.75804145198913279300302692483, −3.12055332533458956793579125277, −1.05052203823584705247662435382,
1.82886814026443451118144403358, 2.60527716356571346616684147980, 3.81125414896706093193247574030, 4.98705371568924039154066464125, 5.79423867943760132831301338289, 6.82514986844470275947347326894, 8.298216817237748530287941402526, 8.981836211447332365194859258066, 9.581430094652173552151662667992, 10.27261965224485812106589295425
