L(s) = 1 | + (−2.5 − 0.866i)7-s + (−1 + 1.73i)11-s + 13-s + (−1 + 1.73i)17-s + (−2.5 − 4.33i)19-s + (3 + 5.19i)23-s + (2.5 − 4.33i)25-s + 8·29-s + (1.5 − 2.59i)31-s + (4.5 + 7.79i)37-s − 2·41-s + 43-s + (4 + 6.92i)47-s + (5.5 + 4.33i)49-s + (3 − 5.19i)53-s + ⋯ |
L(s) = 1 | + (−0.944 − 0.327i)7-s + (−0.301 + 0.522i)11-s + 0.277·13-s + (−0.242 + 0.420i)17-s + (−0.573 − 0.993i)19-s + (0.625 + 1.08i)23-s + (0.5 − 0.866i)25-s + 1.48·29-s + (0.269 − 0.466i)31-s + (0.739 + 1.28i)37-s − 0.312·41-s + 0.152·43-s + (0.583 + 1.01i)47-s + (0.785 + 0.618i)49-s + (0.412 − 0.713i)53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.126i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 - 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.451447220\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.451447220\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (2.5 + 0.866i)T \) |
good | 5 | \( 1 + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (1 - 1.73i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 - T + 13T^{2} \) |
| 17 | \( 1 + (1 - 1.73i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (2.5 + 4.33i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-3 - 5.19i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 8T + 29T^{2} \) |
| 31 | \( 1 + (-1.5 + 2.59i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-4.5 - 7.79i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 - T + 43T^{2} \) |
| 47 | \( 1 + (-4 - 6.92i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-3 + 5.19i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-3 + 5.19i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-1 - 1.73i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-2.5 + 4.33i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 4T + 71T^{2} \) |
| 73 | \( 1 + (-5.5 + 9.52i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-2.5 - 4.33i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + (-6 - 10.3i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 18T + 97T^{2} \) |
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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
−9.210220126892870990936180394985, −8.428174046964265616624622357564, −7.57565832573131320214356719704, −6.63368033627424095526584596572, −6.29429945650867780502041776543, −5.01781905411925733435518817575, −4.30573812498332139594262724586, −3.23645231162302387639856389423, −2.39035303845600298420578278056, −0.835380808261731764761490791627,
0.75548003203376718681735610263, 2.38941155371295873847834442547, 3.18319680120722497899312602920, 4.14319712929519341650642299714, 5.20647901203443763894440876646, 6.04700827487207487213898640207, 6.67866005004160149304255744705, 7.51874296049311745459624114336, 8.729983498780691958254515901069, 8.802887512635859271236806134655