| L(s) = 1 | + (−1.13 + 0.845i)2-s + (−1.00 + 1.41i)3-s + (0.569 − 1.91i)4-s + (−0.908 − 1.35i)5-s + (−0.0523 − 2.44i)6-s + (1.44 − 3.49i)7-s + (0.975 + 2.65i)8-s + (−0.976 − 2.83i)9-s + (2.17 + 0.772i)10-s + (3.39 + 0.675i)11-s + (2.13 + 2.73i)12-s + (−1.72 − 1.15i)13-s + (1.31 + 5.18i)14-s + (2.82 + 0.0863i)15-s + (−3.35 − 2.18i)16-s + (−1.04 − 1.04i)17-s + ⋯ |
| L(s) = 1 | + (−0.801 + 0.597i)2-s + (−0.580 + 0.814i)3-s + (0.284 − 0.958i)4-s + (−0.406 − 0.607i)5-s + (−0.0213 − 0.999i)6-s + (0.547 − 1.32i)7-s + (0.344 + 0.938i)8-s + (−0.325 − 0.945i)9-s + (0.688 + 0.244i)10-s + (1.02 + 0.203i)11-s + (0.614 + 0.788i)12-s + (−0.477 − 0.319i)13-s + (0.351 + 1.38i)14-s + (0.730 + 0.0222i)15-s + (−0.837 − 0.546i)16-s + (−0.254 − 0.254i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.977 + 0.208i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.977 + 0.208i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.641842 - 0.0677019i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.641842 - 0.0677019i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.13 - 0.845i)T \) |
| 3 | \( 1 + (1.00 - 1.41i)T \) |
| good | 5 | \( 1 + (0.908 + 1.35i)T + (-1.91 + 4.61i)T^{2} \) |
| 7 | \( 1 + (-1.44 + 3.49i)T + (-4.94 - 4.94i)T^{2} \) |
| 11 | \( 1 + (-3.39 - 0.675i)T + (10.1 + 4.20i)T^{2} \) |
| 13 | \( 1 + (1.72 + 1.15i)T + (4.97 + 12.0i)T^{2} \) |
| 17 | \( 1 + (1.04 + 1.04i)T + 17iT^{2} \) |
| 19 | \( 1 + (-6.92 - 4.62i)T + (7.27 + 17.5i)T^{2} \) |
| 23 | \( 1 + (1.88 + 4.54i)T + (-16.2 + 16.2i)T^{2} \) |
| 29 | \( 1 + (1.76 + 8.89i)T + (-26.7 + 11.0i)T^{2} \) |
| 31 | \( 1 - 2.37T + 31T^{2} \) |
| 37 | \( 1 + (-1.65 - 2.47i)T + (-14.1 + 34.1i)T^{2} \) |
| 41 | \( 1 + (1.72 - 0.713i)T + (28.9 - 28.9i)T^{2} \) |
| 43 | \( 1 + (9.26 + 1.84i)T + (39.7 + 16.4i)T^{2} \) |
| 47 | \( 1 + (-0.0451 + 0.0451i)T - 47iT^{2} \) |
| 53 | \( 1 + (1.12 - 5.67i)T + (-48.9 - 20.2i)T^{2} \) |
| 59 | \( 1 + (-5.63 + 3.76i)T + (22.5 - 54.5i)T^{2} \) |
| 61 | \( 1 + (-1.08 - 5.44i)T + (-56.3 + 23.3i)T^{2} \) |
| 67 | \( 1 + (8.44 - 1.67i)T + (61.8 - 25.6i)T^{2} \) |
| 71 | \( 1 + (-7.65 - 3.16i)T + (50.2 + 50.2i)T^{2} \) |
| 73 | \( 1 + (-6.47 + 2.68i)T + (51.6 - 51.6i)T^{2} \) |
| 79 | \( 1 + (4.09 - 4.09i)T - 79iT^{2} \) |
| 83 | \( 1 + (3.39 - 5.07i)T + (-31.7 - 76.6i)T^{2} \) |
| 89 | \( 1 + (-2.79 - 1.15i)T + (62.9 + 62.9i)T^{2} \) |
| 97 | \( 1 + 5.24iT - 97T^{2} \) |
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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
−12.02806323803198343483907155457, −11.42144325064952574419715279316, −10.19731812983049488098792000865, −9.745894453877073461460642161170, −8.451023898657878788302263610454, −7.50096527279956004511845225287, −6.33383433421907444687893870225, −4.97453446769181384837549837776, −4.08495357223034891207936126734, −0.880643260618766007130841779543,
1.69468868907055850813838051399, 3.14374212438244846067853782493, 5.19667556765943590977736450945, 6.69071921543710860626420993381, 7.46990897092536089883982844297, 8.621905647052712691617192587711, 9.499225931379243918069991334686, 11.02311030581490034074849864876, 11.66183459826991240613134730400, 11.99682224954819260370755455516
