| L(s) = 1 | + (0.574 − 1.63i)3-s + 5-s − 4.28i·7-s + (−2.33 − 1.87i)9-s − 1.89·11-s + 0.406i·13-s + (0.574 − 1.63i)15-s − 4.17i·17-s − 5.36·19-s + (−7.00 − 2.46i)21-s − 6.34i·23-s + 25-s + (−4.41 + 2.74i)27-s + 3.90i·29-s + 3.07i·31-s + ⋯ |
| L(s) = 1 | + (0.331 − 0.943i)3-s + 0.447·5-s − 1.61i·7-s + (−0.779 − 0.625i)9-s − 0.572·11-s + 0.112i·13-s + (0.148 − 0.421i)15-s − 1.01i·17-s − 1.23·19-s + (−1.52 − 0.537i)21-s − 1.32i·23-s + 0.200·25-s + (−0.849 + 0.528i)27-s + 0.725i·29-s + 0.552i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.847 - 0.530i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.847 - 0.530i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.183156151\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.183156151\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.574 + 1.63i)T \) |
| 5 | \( 1 - T \) |
| 67 | \( 1 + (1.79 - 7.98i)T \) |
| good | 7 | \( 1 + 4.28iT - 7T^{2} \) |
| 11 | \( 1 + 1.89T + 11T^{2} \) |
| 13 | \( 1 - 0.406iT - 13T^{2} \) |
| 17 | \( 1 + 4.17iT - 17T^{2} \) |
| 19 | \( 1 + 5.36T + 19T^{2} \) |
| 23 | \( 1 + 6.34iT - 23T^{2} \) |
| 29 | \( 1 - 3.90iT - 29T^{2} \) |
| 31 | \( 1 - 3.07iT - 31T^{2} \) |
| 37 | \( 1 - 7.35T + 37T^{2} \) |
| 41 | \( 1 + 6.75T + 41T^{2} \) |
| 43 | \( 1 - 7.33iT - 43T^{2} \) |
| 47 | \( 1 + 0.438iT - 47T^{2} \) |
| 53 | \( 1 - 5.23T + 53T^{2} \) |
| 59 | \( 1 + 12.3iT - 59T^{2} \) |
| 61 | \( 1 - 7.71iT - 61T^{2} \) |
| 71 | \( 1 + 2.28iT - 71T^{2} \) |
| 73 | \( 1 - 0.538T + 73T^{2} \) |
| 79 | \( 1 + 1.67iT - 79T^{2} \) |
| 83 | \( 1 - 3.18iT - 83T^{2} \) |
| 89 | \( 1 - 10.7iT - 89T^{2} \) |
| 97 | \( 1 + 7.69iT - 97T^{2} \) |
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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
−8.030893509139564786644895338708, −7.05429518430162069300852114884, −6.86111806445121017392388377206, −6.02755783533327118575500825020, −4.93512286593271997339881138104, −4.21338940573835484348575527721, −3.13901912663764297871199240022, −2.35904216521520604612608927243, −1.26883739838878104430840063395, −0.30882754600175428041815307909,
1.98131048130732055065516577889, 2.48372357393318973342124619995, 3.45568376712641638926382034188, 4.35108492841950147030112508083, 5.28186986147743881514355005402, 5.76792062025823003475425510256, 6.34303019773350246706289474475, 7.71997246428610533907554077057, 8.343918158496034306832045648163, 8.911571840699300626985975481649
