L(s) = 1 | + (−0.5 + 0.866i)2-s + (−0.499 − 0.866i)4-s + (−0.880 + 1.52i)5-s + (2.56 + 0.658i)7-s + 0.999·8-s + (−0.880 − 1.52i)10-s + (−3.06 − 5.30i)11-s + 0.760·13-s + (−1.85 + 1.88i)14-s + (−0.5 + 0.866i)16-s + (−3.42 − 5.92i)17-s + (0.971 − 1.68i)19-s + 1.76·20-s + 6.12·22-s + (0.210 − 0.364i)23-s + ⋯ |
L(s) = 1 | + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s + (−0.393 + 0.681i)5-s + (0.968 + 0.249i)7-s + 0.353·8-s + (−0.278 − 0.482i)10-s + (−0.923 − 1.59i)11-s + 0.211·13-s + (−0.494 + 0.505i)14-s + (−0.125 + 0.216i)16-s + (−0.829 − 1.43i)17-s + (0.222 − 0.385i)19-s + 0.393·20-s + 1.30·22-s + (0.0438 − 0.0760i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.757 + 0.653i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.757 + 0.653i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9510944833\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9510944833\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 - 0.866i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-2.56 - 0.658i)T \) |
good | 5 | \( 1 + (0.880 - 1.52i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (3.06 + 5.30i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 - 0.760T + 13T^{2} \) |
| 17 | \( 1 + (3.42 + 5.92i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.971 + 1.68i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.210 + 0.364i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 1.46T + 29T^{2} \) |
| 31 | \( 1 + (3.85 + 6.67i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-1.44 + 2.49i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 6.94T + 41T^{2} \) |
| 43 | \( 1 + 8.66T + 43T^{2} \) |
| 47 | \( 1 + (0.830 - 1.43i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (0.112 + 0.195i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (0.993 + 1.72i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-5.17 + 8.96i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (3.39 + 5.87i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 10.7T + 71T^{2} \) |
| 73 | \( 1 + (-0.153 - 0.265i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-6.72 + 11.6i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 3.12T + 83T^{2} \) |
| 89 | \( 1 + (-1.30 + 2.25i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 3.63T + 97T^{2} \) |
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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
−9.481284175250142816128854647224, −8.797102956777400203131355396194, −7.938119160753658906747240287174, −7.45166230916158144235024823614, −6.44303003601740765199564735410, −5.50671048075523287646906092385, −4.79829197923494158258656653615, −3.43342364269552035684143509852, −2.35686388516774258797555001776, −0.49050841873831015182161904990,
1.39629836555097936674248722684, 2.27646545032197456526097968464, 3.87694579987052817482251669890, 4.58955964254241696566197830586, 5.31515366485477242459885406262, 6.81553695050589644032417289480, 7.75307271914237347755481972071, 8.288576258260481217283226684880, 9.018721149592545280114631297549, 10.11721233274492699383756854610