| L(s) = 1 | + (−0.978 − 0.207i)2-s + (0.768 + 1.33i)3-s + (0.913 + 0.406i)4-s + (−0.194 − 1.85i)5-s + (−0.475 − 1.46i)6-s + (−1.59 + 2.11i)7-s + (−0.809 − 0.587i)8-s + (0.317 − 0.550i)9-s + (−0.194 + 1.85i)10-s + (−0.378 + 3.60i)11-s + (0.160 + 1.52i)12-s + (−0.767 − 2.36i)13-s + (1.99 − 1.73i)14-s + (2.31 − 1.68i)15-s + (0.669 + 0.743i)16-s + (−0.846 + 8.05i)17-s + ⋯ |
| L(s) = 1 | + (−0.691 − 0.147i)2-s + (0.443 + 0.768i)3-s + (0.456 + 0.203i)4-s + (−0.0870 − 0.828i)5-s + (−0.193 − 0.597i)6-s + (−0.601 + 0.798i)7-s + (−0.286 − 0.207i)8-s + (0.105 − 0.183i)9-s + (−0.0615 + 0.585i)10-s + (−0.114 + 1.08i)11-s + (0.0464 + 0.441i)12-s + (−0.212 − 0.655i)13-s + (0.533 − 0.463i)14-s + (0.598 − 0.434i)15-s + (0.167 + 0.185i)16-s + (−0.205 + 1.95i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 574 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0996 - 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 574 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0996 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.760072 + 0.687779i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.760072 + 0.687779i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.978 + 0.207i)T \) |
| 7 | \( 1 + (1.59 - 2.11i)T \) |
| 41 | \( 1 + (6.38 - 0.447i)T \) |
| good | 3 | \( 1 + (-0.768 - 1.33i)T + (-1.5 + 2.59i)T^{2} \) |
| 5 | \( 1 + (0.194 + 1.85i)T + (-4.89 + 1.03i)T^{2} \) |
| 11 | \( 1 + (0.378 - 3.60i)T + (-10.7 - 2.28i)T^{2} \) |
| 13 | \( 1 + (0.767 + 2.36i)T + (-10.5 + 7.64i)T^{2} \) |
| 17 | \( 1 + (0.846 - 8.05i)T + (-16.6 - 3.53i)T^{2} \) |
| 19 | \( 1 + (-5.24 - 5.82i)T + (-1.98 + 18.8i)T^{2} \) |
| 23 | \( 1 + (0.889 + 0.189i)T + (21.0 + 9.35i)T^{2} \) |
| 29 | \( 1 + (3.82 - 2.78i)T + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (-0.467 + 4.44i)T + (-30.3 - 6.44i)T^{2} \) |
| 37 | \( 1 + (-0.506 - 4.82i)T + (-36.1 + 7.69i)T^{2} \) |
| 43 | \( 1 + (-1.93 - 5.96i)T + (-34.7 + 25.2i)T^{2} \) |
| 47 | \( 1 + (-3.06 - 0.650i)T + (42.9 + 19.1i)T^{2} \) |
| 53 | \( 1 + (-0.988 - 0.440i)T + (35.4 + 39.3i)T^{2} \) |
| 59 | \( 1 + (1.04 - 1.15i)T + (-6.16 - 58.6i)T^{2} \) |
| 61 | \( 1 + (4.16 + 4.63i)T + (-6.37 + 60.6i)T^{2} \) |
| 67 | \( 1 + (4.49 + 2.00i)T + (44.8 + 49.7i)T^{2} \) |
| 71 | \( 1 + (-1.86 - 1.35i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (-6.13 - 10.6i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-4.01 + 6.94i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 5.55T + 83T^{2} \) |
| 89 | \( 1 + (-2.22 - 2.47i)T + (-9.30 + 88.5i)T^{2} \) |
| 97 | \( 1 + (-11.0 + 8.03i)T + (29.9 - 92.2i)T^{2} \) |
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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
−10.47094874752104950787249049833, −9.898867227750014648250972003206, −9.310848074384837224866567482055, −8.479242463848819749366987927888, −7.77071271281506706043045501800, −6.40531550473897233633371688081, −5.37797169872948547779885489756, −4.14172022637114404551940785040, −3.15433226707343408665448901380, −1.63142656999771830994727197311,
0.71256771702378703907056488151, 2.49676869135771051088626012068, 3.30528705007765330405355582777, 5.04173209966437297256011974052, 6.50369237325804985003919930788, 7.29717875608018317341541591150, 7.40128819845575137307492990590, 8.829407398615088904775360018771, 9.496594566259020978430356683583, 10.54510974424066988710469379617
