| L(s) = 1 | − 0.0872·2-s − 3-s − 1.99·4-s − 0.477·5-s + 0.0872·6-s + 0.435·7-s + 0.348·8-s + 9-s + 0.0416·10-s − 11-s + 1.99·12-s + 13-s − 0.0380·14-s + 0.477·15-s + 3.95·16-s + 3.29·17-s − 0.0872·18-s + 4.43·19-s + 0.950·20-s − 0.435·21-s + 0.0872·22-s + 3.16·23-s − 0.348·24-s − 4.77·25-s − 0.0872·26-s − 27-s − 0.867·28-s + ⋯ |
| L(s) = 1 | − 0.0616·2-s − 0.577·3-s − 0.996·4-s − 0.213·5-s + 0.0356·6-s + 0.164·7-s + 0.123·8-s + 0.333·9-s + 0.0131·10-s − 0.301·11-s + 0.575·12-s + 0.277·13-s − 0.0101·14-s + 0.123·15-s + 0.988·16-s + 0.799·17-s − 0.0205·18-s + 1.01·19-s + 0.212·20-s − 0.0950·21-s + 0.0186·22-s + 0.660·23-s − 0.0710·24-s − 0.954·25-s − 0.0171·26-s − 0.192·27-s − 0.164·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8786522081\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8786522081\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| good | 2 | \( 1 + 0.0872T + 2T^{2} \) |
| 5 | \( 1 + 0.477T + 5T^{2} \) |
| 7 | \( 1 - 0.435T + 7T^{2} \) |
| 17 | \( 1 - 3.29T + 17T^{2} \) |
| 19 | \( 1 - 4.43T + 19T^{2} \) |
| 23 | \( 1 - 3.16T + 23T^{2} \) |
| 29 | \( 1 - 6.02T + 29T^{2} \) |
| 31 | \( 1 - 10.6T + 31T^{2} \) |
| 37 | \( 1 - 3.42T + 37T^{2} \) |
| 41 | \( 1 + 7.54T + 41T^{2} \) |
| 43 | \( 1 - 12.4T + 43T^{2} \) |
| 47 | \( 1 + 3.25T + 47T^{2} \) |
| 53 | \( 1 + 9.81T + 53T^{2} \) |
| 59 | \( 1 + 12.9T + 59T^{2} \) |
| 61 | \( 1 - 10.3T + 61T^{2} \) |
| 67 | \( 1 - 7.50T + 67T^{2} \) |
| 71 | \( 1 - 14.5T + 71T^{2} \) |
| 73 | \( 1 - 0.344T + 73T^{2} \) |
| 79 | \( 1 + 7.55T + 79T^{2} \) |
| 83 | \( 1 + 9.03T + 83T^{2} \) |
| 89 | \( 1 - 5.90T + 89T^{2} \) |
| 97 | \( 1 - 12.8T + 97T^{2} \) |
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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
−11.20734284248916712220042504578, −10.10766950988103630771935242325, −9.559320571851867634632311795629, −8.322277979667446718764328423230, −7.68476205059694681298338172017, −6.31900800511209078782859840678, −5.27368577745143377462680955091, −4.49469340812319114559957848267, −3.22439467580610903491995591848, −0.974181413234358083181963502538,
0.974181413234358083181963502538, 3.22439467580610903491995591848, 4.49469340812319114559957848267, 5.27368577745143377462680955091, 6.31900800511209078782859840678, 7.68476205059694681298338172017, 8.322277979667446718764328423230, 9.559320571851867634632311795629, 10.10766950988103630771935242325, 11.20734284248916712220042504578
