L(s) = 1 | + (1.33 − 0.462i)2-s + (1.57 − 1.23i)4-s + (1.56 + 0.902i)5-s + (2.63 − 0.217i)7-s + (1.52 − 2.38i)8-s + (2.50 + 0.482i)10-s + (−4.48 + 2.58i)11-s + 0.840i·13-s + (3.42 − 1.51i)14-s + (0.938 − 3.88i)16-s + (−2.45 − 4.25i)17-s + (4.87 + 2.81i)19-s + (3.57 − 0.515i)20-s + (−4.78 + 5.53i)22-s + (−3.05 + 5.28i)23-s + ⋯ |
L(s) = 1 | + (0.944 − 0.327i)2-s + (0.785 − 0.618i)4-s + (0.698 + 0.403i)5-s + (0.996 − 0.0820i)7-s + (0.539 − 0.841i)8-s + (0.792 + 0.152i)10-s + (−1.35 + 0.779i)11-s + 0.232i·13-s + (0.914 − 0.403i)14-s + (0.234 − 0.972i)16-s + (−0.595 − 1.03i)17-s + (1.11 + 0.646i)19-s + (0.798 − 0.115i)20-s + (−1.02 + 1.17i)22-s + (−0.636 + 1.10i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.911 + 0.412i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.911 + 0.412i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.79949 - 0.603737i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.79949 - 0.603737i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.33 + 0.462i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-2.63 + 0.217i)T \) |
good | 5 | \( 1 + (-1.56 - 0.902i)T + (2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (4.48 - 2.58i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 - 0.840iT - 13T^{2} \) |
| 17 | \( 1 + (2.45 + 4.25i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-4.87 - 2.81i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (3.05 - 5.28i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 0.439iT - 29T^{2} \) |
| 31 | \( 1 + (3.66 + 6.35i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (4.56 + 2.63i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 6.23T + 41T^{2} \) |
| 43 | \( 1 - 7.34iT - 43T^{2} \) |
| 47 | \( 1 + (-2.83 + 4.91i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (1.15 - 0.669i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (7.31 - 4.22i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (4.77 + 2.75i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (0.647 - 0.373i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 9.08T + 71T^{2} \) |
| 73 | \( 1 + (-3.70 - 6.42i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (8.68 - 15.0i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 1.45iT - 83T^{2} \) |
| 89 | \( 1 + (-3.10 + 5.37i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 5.81T + 97T^{2} \) |
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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
−11.02812297485791867081658896249, −10.11610272728417700617051014683, −9.487650429130338858835676327575, −7.75653966317558231779417996530, −7.25629802137547923612255404720, −5.84389908019395673541541763180, −5.21750030367537159657595705416, −4.21577642416076668795107471817, −2.70180552674419397073425057419, −1.81545362740944682699513535066,
1.85464114777418517004879940113, 3.07275756870055996435890420822, 4.55645784152395447196810271489, 5.35798570940357148187378554735, 5.98356782505281117566287366975, 7.32301184851435165010043414050, 8.177804617102215127961711906065, 8.916338001536873926066808492238, 10.53886758973750747299555032339, 10.90187991354765864473907001388