| L(s) = 1 | + (−1.49 − 0.306i)2-s + (−0.987 + 0.160i)3-s + (0.315 + 0.134i)4-s + (−0.217 + 0.880i)5-s + (1.52 + 0.0616i)6-s + (0.786 − 0.373i)7-s + (2.08 + 1.44i)8-s + (0.948 − 0.316i)9-s + (0.595 − 1.25i)10-s + (−0.216 + 0.649i)11-s + (−0.333 − 0.0821i)12-s + (−2.69 − 2.39i)13-s + (−1.29 + 0.318i)14-s + (0.0729 − 0.904i)15-s + (−3.16 − 3.29i)16-s + (−1.35 − 2.85i)17-s + ⋯ |
| L(s) = 1 | + (−1.06 − 0.216i)2-s + (−0.569 + 0.0926i)3-s + (0.157 + 0.0672i)4-s + (−0.0970 + 0.393i)5-s + (0.624 + 0.0251i)6-s + (0.297 − 0.141i)7-s + (0.737 + 0.509i)8-s + (0.316 − 0.105i)9-s + (0.188 − 0.396i)10-s + (−0.0653 + 0.195i)11-s + (−0.0961 − 0.0237i)12-s + (−0.747 − 0.664i)13-s + (−0.345 + 0.0852i)14-s + (0.0188 − 0.233i)15-s + (−0.791 − 0.823i)16-s + (−0.328 − 0.692i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.00660 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.00660 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.280049 + 0.278207i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.280049 + 0.278207i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.987 - 0.160i)T \) |
| 13 | \( 1 + (2.69 + 2.39i)T \) |
| good | 2 | \( 1 + (1.49 + 0.306i)T + (1.83 + 0.783i)T^{2} \) |
| 5 | \( 1 + (0.217 - 0.880i)T + (-4.42 - 2.32i)T^{2} \) |
| 7 | \( 1 + (-0.786 + 0.373i)T + (4.42 - 5.42i)T^{2} \) |
| 11 | \( 1 + (0.216 - 0.649i)T + (-8.79 - 6.60i)T^{2} \) |
| 17 | \( 1 + (1.35 + 2.85i)T + (-10.7 + 13.1i)T^{2} \) |
| 19 | \( 1 + (2.27 + 1.31i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-4.63 - 8.03i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (1.59 - 7.79i)T + (-26.6 - 11.3i)T^{2} \) |
| 31 | \( 1 + (0.146 + 0.279i)T + (-17.6 + 25.5i)T^{2} \) |
| 37 | \( 1 + (2.64 - 4.17i)T + (-15.8 - 33.4i)T^{2} \) |
| 41 | \( 1 + (-1.21 - 7.50i)T + (-38.8 + 12.9i)T^{2} \) |
| 43 | \( 1 + (-0.806 + 0.509i)T + (18.4 - 38.8i)T^{2} \) |
| 47 | \( 1 + (6.00 + 0.729i)T + (45.6 + 11.2i)T^{2} \) |
| 53 | \( 1 + (-0.296 + 0.429i)T + (-18.7 - 49.5i)T^{2} \) |
| 59 | \( 1 + (-9.80 - 9.41i)T + (2.37 + 58.9i)T^{2} \) |
| 61 | \( 1 + (6.60 - 0.533i)T + (60.2 - 9.78i)T^{2} \) |
| 67 | \( 1 + (-3.39 - 7.97i)T + (-46.4 + 48.3i)T^{2} \) |
| 71 | \( 1 + (2.47 - 2.01i)T + (14.2 - 69.5i)T^{2} \) |
| 73 | \( 1 + (-0.513 - 0.579i)T + (-8.79 + 72.4i)T^{2} \) |
| 79 | \( 1 + (-1.41 + 11.6i)T + (-76.7 - 18.9i)T^{2} \) |
| 83 | \( 1 + (8.68 - 3.29i)T + (62.1 - 55.0i)T^{2} \) |
| 89 | \( 1 + (15.1 - 8.76i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-6.54 - 1.89i)T + (81.9 + 51.8i)T^{2} \) |
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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
−11.06148253230288510889074854617, −10.20557111504420495135124161221, −9.491846014357837386838377905160, −8.622833399768047662571945570219, −7.47618263449792179662531051859, −6.97975246737321726728991310089, −5.36089345202551334159299492716, −4.69784480453281026112933470232, −2.97611768624866098963682162224, −1.32425376838010657222346978536,
0.39818713400046121373672270677, 2.03353827764624311497319434854, 4.16607646996988642092205293369, 4.94055275623861757998891481896, 6.34956504145149565844975292033, 7.13849310854969862400558733338, 8.245683282390120940595673821222, 8.758770407480806407229986648436, 9.745419141935409848482680903722, 10.57681794001406702959600598533
