L(s) = 1 | + (0.309 + 0.951i)2-s + (2.28 − 1.66i)3-s + (−0.809 + 0.587i)4-s + (0.312 − 2.21i)5-s + (2.28 + 1.66i)6-s − 5.10·7-s + (−0.809 − 0.587i)8-s + (1.53 − 4.73i)9-s + (2.20 − 0.386i)10-s + (−0.334 − 1.02i)11-s + (−0.872 + 2.68i)12-s + (0.0738 − 0.227i)13-s + (−1.57 − 4.85i)14-s + (−2.96 − 5.57i)15-s + (0.309 − 0.951i)16-s + (−3.12 − 2.26i)17-s + ⋯ |
L(s) = 1 | + (0.218 + 0.672i)2-s + (1.31 − 0.958i)3-s + (−0.404 + 0.293i)4-s + (0.139 − 0.990i)5-s + (0.933 + 0.677i)6-s − 1.92·7-s + (−0.286 − 0.207i)8-s + (0.512 − 1.57i)9-s + (0.696 − 0.122i)10-s + (−0.100 − 0.310i)11-s + (−0.251 + 0.775i)12-s + (0.0204 − 0.0630i)13-s + (−0.421 − 1.29i)14-s + (−0.764 − 1.44i)15-s + (0.0772 − 0.237i)16-s + (−0.757 − 0.550i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.216 + 0.976i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.216 + 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.07543 - 1.33978i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.07543 - 1.33978i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-0.312 + 2.21i)T \) |
| 19 | \( 1 + (0.809 + 0.587i)T \) |
good | 3 | \( 1 + (-2.28 + 1.66i)T + (0.927 - 2.85i)T^{2} \) |
| 7 | \( 1 + 5.10T + 7T^{2} \) |
| 11 | \( 1 + (0.334 + 1.02i)T + (-8.89 + 6.46i)T^{2} \) |
| 13 | \( 1 + (-0.0738 + 0.227i)T + (-10.5 - 7.64i)T^{2} \) |
| 17 | \( 1 + (3.12 + 2.26i)T + (5.25 + 16.1i)T^{2} \) |
| 23 | \( 1 + (0.561 + 1.72i)T + (-18.6 + 13.5i)T^{2} \) |
| 29 | \( 1 + (-4.48 + 3.25i)T + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (-6.99 - 5.08i)T + (9.57 + 29.4i)T^{2} \) |
| 37 | \( 1 + (0.0153 - 0.0471i)T + (-29.9 - 21.7i)T^{2} \) |
| 41 | \( 1 + (-2.80 + 8.61i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 + 8.06T + 43T^{2} \) |
| 47 | \( 1 + (-4.02 + 2.92i)T + (14.5 - 44.6i)T^{2} \) |
| 53 | \( 1 + (0.679 - 0.494i)T + (16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (-2.92 + 8.98i)T + (-47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + (-2.32 - 7.14i)T + (-49.3 + 35.8i)T^{2} \) |
| 67 | \( 1 + (-6.80 - 4.94i)T + (20.7 + 63.7i)T^{2} \) |
| 71 | \( 1 + (5.65 - 4.10i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (0.0228 + 0.0703i)T + (-59.0 + 42.9i)T^{2} \) |
| 79 | \( 1 + (5.45 - 3.96i)T + (24.4 - 75.1i)T^{2} \) |
| 83 | \( 1 + (-1.97 - 1.43i)T + (25.6 + 78.9i)T^{2} \) |
| 89 | \( 1 + (1.92 + 5.91i)T + (-72.0 + 52.3i)T^{2} \) |
| 97 | \( 1 + (-6.09 + 4.43i)T + (29.9 - 92.2i)T^{2} \) |
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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
−9.464677508523446335820416346221, −8.662012935535890387406722546396, −8.364096193240347776181053316093, −7.09428516655649509150831407687, −6.65983835644319973720236500214, −5.72834570386374240631334613492, −4.34616501455670798365291596777, −3.28934647059330078009809678722, −2.43232223966681384998181583774, −0.60156706170086658505083138912,
2.34319122739516183414719099865, 3.01024233543979042316664118072, 3.64835538315095744723450937541, 4.48985481564789036707628544745, 6.07879840424955684158322499721, 6.77074730368744983036010261385, 8.002480021317477955121870459665, 8.996325434029999924568309614246, 9.679449533188916811018506198889, 10.10428091106597279490015999703