L(s) = 1 | + (−1.40 − 1.01i)3-s + (−0.5 − 0.866i)5-s + (1.40 − 2.42i)7-s + (0.936 + 2.85i)9-s + (1.22 − 2.12i)11-s + (−2.43 − 4.22i)13-s + (−0.178 + 1.72i)15-s − 6.87·17-s + 7.34·19-s + (−4.43 + 1.98i)21-s + (1.40 + 2.42i)23-s + (−0.499 + 0.866i)25-s + (1.58 − 4.94i)27-s + (3.37 − 5.84i)29-s + (−1.93 − 3.35i)31-s + ⋯ |
L(s) = 1 | + (−0.809 − 0.586i)3-s + (−0.223 − 0.387i)5-s + (0.530 − 0.918i)7-s + (0.312 + 0.950i)9-s + (0.369 − 0.639i)11-s + (−0.675 − 1.17i)13-s + (−0.0460 + 0.444i)15-s − 1.66·17-s + 1.68·19-s + (−0.968 + 0.432i)21-s + (0.292 + 0.506i)23-s + (−0.0999 + 0.173i)25-s + (0.304 − 0.952i)27-s + (0.626 − 1.08i)29-s + (−0.347 − 0.602i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.989 + 0.142i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.989 + 0.142i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8351384050\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8351384050\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.40 + 1.01i)T \) |
| 5 | \( 1 + (0.5 + 0.866i)T \) |
good | 7 | \( 1 + (-1.40 + 2.42i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-1.22 + 2.12i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (2.43 + 4.22i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + 6.87T + 17T^{2} \) |
| 19 | \( 1 - 7.34T + 19T^{2} \) |
| 23 | \( 1 + (-1.40 - 2.42i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-3.37 + 5.84i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (1.93 + 3.35i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 0.872T + 37T^{2} \) |
| 41 | \( 1 + (-1.06 - 1.84i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (1.22 - 2.12i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (2.98 - 5.16i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + 4T + 53T^{2} \) |
| 59 | \( 1 + (2.80 + 4.85i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (2.93 - 5.08i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (7.88 + 13.6i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 3.16T + 71T^{2} \) |
| 73 | \( 1 + 9.74T + 73T^{2} \) |
| 79 | \( 1 + (-6.32 + 10.9i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (7.72 - 13.3i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + 16.7T + 89T^{2} \) |
| 97 | \( 1 + (-0.872 + 1.51i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−9.182048015822620702617187251325, −7.908201665895648478964772408436, −7.70836125282040533973573267947, −6.73406180908099524704797243848, −5.82709182469170528506952524136, −4.96430438558849488360793548154, −4.26898897863992987524918820103, −2.89809705435619363359998763902, −1.38790881880210903600317056402, −0.39602432871546571969236509593,
1.69412408142830060879929082305, 2.93057643499232815587457039140, 4.26609597171395673690221590824, 4.83926036906268329741570878222, 5.65587611271998591290334038676, 6.91722324472816678289560934195, 7.00132679278407548533737647905, 8.590799691565445576490462673449, 9.179531148735170430419187798610, 9.856721739279314554273598591804