L(s) = 1 | + (0.309 + 0.951i)2-s + (−0.809 + 0.587i)4-s + (−1.80 + 1.31i)5-s − 3·7-s + (−0.809 − 0.587i)8-s + (−1.80 − 1.31i)10-s + (−1.30 − 4.02i)11-s + (0.309 − 0.951i)13-s + (−0.927 − 2.85i)14-s + (0.309 − 0.951i)16-s + (0.927 + 0.673i)17-s + (−4.73 − 3.44i)19-s + (0.690 − 2.12i)20-s + (3.42 − 2.48i)22-s + (0.545 + 1.67i)23-s + ⋯ |
L(s) = 1 | + (0.218 + 0.672i)2-s + (−0.404 + 0.293i)4-s + (−0.809 + 0.587i)5-s − 1.13·7-s + (−0.286 − 0.207i)8-s + (−0.572 − 0.415i)10-s + (−0.394 − 1.21i)11-s + (0.0857 − 0.263i)13-s + (−0.247 − 0.762i)14-s + (0.0772 − 0.237i)16-s + (0.224 + 0.163i)17-s + (−1.08 − 0.789i)19-s + (0.154 − 0.475i)20-s + (0.730 − 0.530i)22-s + (0.113 + 0.349i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.535 + 0.844i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.535 + 0.844i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 + (1.80 - 1.31i)T \) |
good | 7 | \( 1 + 3T + 7T^{2} \) |
| 11 | \( 1 + (1.30 + 4.02i)T + (-8.89 + 6.46i)T^{2} \) |
| 13 | \( 1 + (-0.309 + 0.951i)T + (-10.5 - 7.64i)T^{2} \) |
| 17 | \( 1 + (-0.927 - 0.673i)T + (5.25 + 16.1i)T^{2} \) |
| 19 | \( 1 + (4.73 + 3.44i)T + (5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 + (-0.545 - 1.67i)T + (-18.6 + 13.5i)T^{2} \) |
| 29 | \( 1 + (7.66 - 5.56i)T + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (-0.190 - 0.138i)T + (9.57 + 29.4i)T^{2} \) |
| 37 | \( 1 + (2.57 - 7.91i)T + (-29.9 - 21.7i)T^{2} \) |
| 41 | \( 1 + (0.454 - 1.40i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 + 6.23T + 43T^{2} \) |
| 47 | \( 1 + (9.66 - 7.02i)T + (14.5 - 44.6i)T^{2} \) |
| 53 | \( 1 + (-8.47 + 6.15i)T + (16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (-1.38 + 4.25i)T + (-47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + (2.73 + 8.42i)T + (-49.3 + 35.8i)T^{2} \) |
| 67 | \( 1 + (-8.28 - 6.01i)T + (20.7 + 63.7i)T^{2} \) |
| 71 | \( 1 + (2.42 - 1.76i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (-2.38 - 7.33i)T + (-59.0 + 42.9i)T^{2} \) |
| 79 | \( 1 + (-5.85 + 4.25i)T + (24.4 - 75.1i)T^{2} \) |
| 83 | \( 1 + (3.66 + 2.66i)T + (25.6 + 78.9i)T^{2} \) |
| 89 | \( 1 + (1.38 + 4.25i)T + (-72.0 + 52.3i)T^{2} \) |
| 97 | \( 1 + (7.73 - 5.62i)T + (29.9 - 92.2i)T^{2} \) |
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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
−10.86161890619523920080965326172, −9.865085377764723321942026647664, −8.717598904629978589190046805794, −8.014113829548500310170152011149, −6.88384144510362469895364221631, −6.31584176006740511372226371408, −5.14148103799231013348664421939, −3.67781629527325832773666766864, −3.05000763854884951145394741735, 0,
2.07865427878540105637347397105, 3.58751192218407239680299447081, 4.34500215480399794681566777645, 5.52780472724710961198477659242, 6.78583686158399291084737199917, 7.80603466757305555041438715132, 8.890526933757651532436505833656, 9.723841942380523264571083336675, 10.45628613905644049944504981219