L(s) = 1 | + (0.309 + 0.951i)2-s + (0.786 − 0.571i)3-s + (−0.809 + 0.587i)4-s + (−2.18 − 0.487i)5-s + (0.786 + 0.571i)6-s + 4.32·7-s + (−0.809 − 0.587i)8-s + (−0.634 + 1.95i)9-s + (−0.210 − 2.22i)10-s + (−0.386 − 1.19i)11-s + (−0.300 + 0.924i)12-s + (−0.691 + 2.12i)13-s + (1.33 + 4.11i)14-s + (−1.99 + 0.863i)15-s + (0.309 − 0.951i)16-s + (2.81 + 2.04i)17-s + ⋯ |
L(s) = 1 | + (0.218 + 0.672i)2-s + (0.454 − 0.329i)3-s + (−0.404 + 0.293i)4-s + (−0.975 − 0.217i)5-s + (0.321 + 0.233i)6-s + 1.63·7-s + (−0.286 − 0.207i)8-s + (−0.211 + 0.651i)9-s + (−0.0666 − 0.703i)10-s + (−0.116 − 0.359i)11-s + (−0.0867 + 0.266i)12-s + (−0.191 + 0.590i)13-s + (0.357 + 1.09i)14-s + (−0.515 + 0.223i)15-s + (0.0772 − 0.237i)16-s + (0.682 + 0.496i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.367 - 0.929i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.367 - 0.929i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.58815 + 1.07977i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.58815 + 1.07977i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.309 - 0.951i)T \) |
| 5 | \( 1 + (2.18 + 0.487i)T \) |
| 19 | \( 1 + (0.809 + 0.587i)T \) |
good | 3 | \( 1 + (-0.786 + 0.571i)T + (0.927 - 2.85i)T^{2} \) |
| 7 | \( 1 - 4.32T + 7T^{2} \) |
| 11 | \( 1 + (0.386 + 1.19i)T + (-8.89 + 6.46i)T^{2} \) |
| 13 | \( 1 + (0.691 - 2.12i)T + (-10.5 - 7.64i)T^{2} \) |
| 17 | \( 1 + (-2.81 - 2.04i)T + (5.25 + 16.1i)T^{2} \) |
| 23 | \( 1 + (-1.43 - 4.42i)T + (-18.6 + 13.5i)T^{2} \) |
| 29 | \( 1 + (-7.07 + 5.13i)T + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (-4.43 - 3.22i)T + (9.57 + 29.4i)T^{2} \) |
| 37 | \( 1 + (0.465 - 1.43i)T + (-29.9 - 21.7i)T^{2} \) |
| 41 | \( 1 + (-1.61 + 4.97i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 + 0.0742T + 43T^{2} \) |
| 47 | \( 1 + (-1.71 + 1.24i)T + (14.5 - 44.6i)T^{2} \) |
| 53 | \( 1 + (5.68 - 4.13i)T + (16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (3.35 - 10.3i)T + (-47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + (-1.07 - 3.31i)T + (-49.3 + 35.8i)T^{2} \) |
| 67 | \( 1 + (2.66 + 1.93i)T + (20.7 + 63.7i)T^{2} \) |
| 71 | \( 1 + (7.96 - 5.78i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (-3.51 - 10.8i)T + (-59.0 + 42.9i)T^{2} \) |
| 79 | \( 1 + (-2.09 + 1.52i)T + (24.4 - 75.1i)T^{2} \) |
| 83 | \( 1 + (9.64 + 7.00i)T + (25.6 + 78.9i)T^{2} \) |
| 89 | \( 1 + (3.75 + 11.5i)T + (-72.0 + 52.3i)T^{2} \) |
| 97 | \( 1 + (-12.5 + 9.13i)T + (29.9 - 92.2i)T^{2} \) |
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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
−10.25352433709857454644530549616, −8.722313162918282260005124763838, −8.436310586997046286888664892915, −7.68606726286853336283853219077, −7.20458375766289615157344250691, −5.78225942169684839838478494981, −4.84175106093296538756611439058, −4.24484078963066264036500042173, −2.88794096724951959307033900710, −1.40079343600267738751393125510,
0.942067768260350148106911794581, 2.55938339586693954861242194967, 3.44368966288929545429523012340, 4.54204694293516351532799092367, 4.98783922131042794183613852194, 6.46273644014782218876252365444, 7.73581848745207478012144816967, 8.210809827551602332498696308317, 8.995402156745678123192192604271, 10.08149345258985294912271741999