| L(s) = 1 | + (−0.309 − 0.951i)2-s + (−0.978 + 0.207i)3-s + (−0.809 + 0.587i)4-s + (−0.5 + 0.866i)5-s + (0.5 + 0.866i)6-s + (−3.16 − 1.40i)7-s + (0.809 + 0.587i)8-s + (0.913 − 0.406i)9-s + (0.978 + 0.207i)10-s + (−0.570 − 5.42i)11-s + (0.669 − 0.743i)12-s + (1.50 + 1.67i)13-s + (−0.361 + 3.44i)14-s + (0.309 − 0.951i)15-s + (0.309 − 0.951i)16-s + (−0.212 + 2.01i)17-s + ⋯ |
| L(s) = 1 | + (−0.218 − 0.672i)2-s + (−0.564 + 0.120i)3-s + (−0.404 + 0.293i)4-s + (−0.223 + 0.387i)5-s + (0.204 + 0.353i)6-s + (−1.19 − 0.532i)7-s + (0.286 + 0.207i)8-s + (0.304 − 0.135i)9-s + (0.309 + 0.0657i)10-s + (−0.172 − 1.63i)11-s + (0.193 − 0.214i)12-s + (0.418 + 0.464i)13-s + (−0.0967 + 0.920i)14-s + (0.0797 − 0.245i)15-s + (0.0772 − 0.237i)16-s + (−0.0514 + 0.489i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.573 - 0.819i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.573 - 0.819i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.452739 + 0.235740i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.452739 + 0.235740i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.309 + 0.951i)T \) |
| 3 | \( 1 + (0.978 - 0.207i)T \) |
| 5 | \( 1 + (0.5 - 0.866i)T \) |
| 31 | \( 1 + (3.20 - 4.55i)T \) |
| good | 7 | \( 1 + (3.16 + 1.40i)T + (4.68 + 5.20i)T^{2} \) |
| 11 | \( 1 + (0.570 + 5.42i)T + (-10.7 + 2.28i)T^{2} \) |
| 13 | \( 1 + (-1.50 - 1.67i)T + (-1.35 + 12.9i)T^{2} \) |
| 17 | \( 1 + (0.212 - 2.01i)T + (-16.6 - 3.53i)T^{2} \) |
| 19 | \( 1 + (-0.391 + 0.434i)T + (-1.98 - 18.8i)T^{2} \) |
| 23 | \( 1 + (0.132 + 0.0959i)T + (7.10 + 21.8i)T^{2} \) |
| 29 | \( 1 + (-2.63 - 8.10i)T + (-23.4 + 17.0i)T^{2} \) |
| 37 | \( 1 + (-2.99 - 5.19i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-4.96 - 1.05i)T + (37.4 + 16.6i)T^{2} \) |
| 43 | \( 1 + (4.53 - 5.04i)T + (-4.49 - 42.7i)T^{2} \) |
| 47 | \( 1 + (3.89 - 11.9i)T + (-38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (3.33 - 1.48i)T + (35.4 - 39.3i)T^{2} \) |
| 59 | \( 1 + (-4.88 + 1.03i)T + (53.8 - 23.9i)T^{2} \) |
| 61 | \( 1 + 11.5T + 61T^{2} \) |
| 67 | \( 1 + (4.21 - 7.30i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-8.30 + 3.69i)T + (47.5 - 52.7i)T^{2} \) |
| 73 | \( 1 + (0.249 + 2.37i)T + (-71.4 + 15.1i)T^{2} \) |
| 79 | \( 1 + (-0.832 + 7.91i)T + (-77.2 - 16.4i)T^{2} \) |
| 83 | \( 1 + (-16.3 - 3.46i)T + (75.8 + 33.7i)T^{2} \) |
| 89 | \( 1 + (-9.90 + 7.19i)T + (27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (2.92 - 2.12i)T + (29.9 - 92.2i)T^{2} \) |
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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
−10.48834908580852663394007778087, −9.486072632967331024342230408110, −8.743348344411553216387365747264, −7.73542621249200776311771507268, −6.56812025652472930179622409410, −6.09862388379993205767198163250, −4.75524412615956325353002858719, −3.52877859463739494146439858522, −3.09685498224988110000117435953, −1.11704862267825330930854308277,
0.32784402573034384239821893521, 2.22589262246928209144270900964, 3.81915791403886957463161870606, 4.85771912993208510907027221989, 5.72956752746394157032585845407, 6.51199259834252673587998069864, 7.32662463264401872126965549146, 8.097521592840427070512002050611, 9.307797164631959720176293185358, 9.694412407464931600698825466561
