L(s) = 1 | + (−3.40 − 1.96i)5-s + (−0.961 + 0.555i)7-s + (1.63 + 2.82i)11-s + (0.124 − 0.216i)13-s + 5.86i·17-s − 2.19i·19-s + (2.79 − 4.83i)23-s + (5.24 + 9.09i)25-s + (−2.35 + 1.36i)29-s + (8.96 + 5.17i)31-s + 4.36·35-s − 0.333·37-s + (5.28 + 3.05i)41-s + (8.50 − 4.91i)43-s + (4.70 + 8.15i)47-s + ⋯ |
L(s) = 1 | + (−1.52 − 0.880i)5-s + (−0.363 + 0.209i)7-s + (0.492 + 0.852i)11-s + (0.0346 − 0.0600i)13-s + 1.42i·17-s − 0.504i·19-s + (0.582 − 1.00i)23-s + (1.04 + 1.81i)25-s + (−0.437 + 0.252i)29-s + (1.61 + 0.930i)31-s + 0.738·35-s − 0.0547·37-s + (0.825 + 0.476i)41-s + (1.29 − 0.749i)43-s + (0.686 + 1.18i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.665 - 0.745i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.665 - 0.745i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.867855 + 0.388586i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.867855 + 0.388586i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (3.40 + 1.96i)T + (2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 + (0.961 - 0.555i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-1.63 - 2.82i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.124 + 0.216i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 - 5.86iT - 17T^{2} \) |
| 19 | \( 1 + 2.19iT - 19T^{2} \) |
| 23 | \( 1 + (-2.79 + 4.83i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (2.35 - 1.36i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-8.96 - 5.17i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 0.333T + 37T^{2} \) |
| 41 | \( 1 + (-5.28 - 3.05i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-8.50 + 4.91i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-4.70 - 8.15i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 - 4.75iT - 53T^{2} \) |
| 59 | \( 1 + (3.26 - 5.65i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-1.07 - 1.85i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (0.501 + 0.289i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 3.26T + 71T^{2} \) |
| 73 | \( 1 + 12.6T + 73T^{2} \) |
| 79 | \( 1 + (7.67 - 4.42i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (2.34 + 4.05i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + 4.75iT - 89T^{2} \) |
| 97 | \( 1 + (-0.916 - 1.58i)T + (-48.5 + 84.0i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.37276814954999839369724147951, −9.149160834271463776980953465207, −8.658250435699490743819370507982, −7.78905588463433719258734991850, −7.00097270423045196371052788745, −5.94712676674339943047870950968, −4.53465301261639225034551350548, −4.23901092349521762732918531412, −2.93201078668759696437388324793, −1.13759230099744394913054573584,
0.56696938341494541877386683937, 2.81363383014867272299183005087, 3.57417231934171119958114318252, 4.40501625787711493214684535121, 5.79712640889674251255563868851, 6.82336751165519668712006082383, 7.45253990771435622980015144930, 8.188956481942846506137467599055, 9.213119194647548708277840307424, 10.10766747035929979537817854605