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
\(L(1)\) |
\(\approx\) |
\(2.082485058\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.082485058\) |
\(L(\frac{3}{2})\) |
|
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
−8.896128638598400707479821494544, −8.380300615545283683005214063971, −7.88331914244165383674418672591, −6.63327296516652306383051610150, −6.09664405674657695851198621875, −5.07129258196558301496825491902, −4.35423131221347129320641975682, −3.30444471021817680671187492401, −2.18051935450950008292518745423, −1.05182543295580171154941226554,
0.997018932130729520798242311801, 2.09987401460661292316185846399, 3.28961494686958588471894368565, 4.37059897616797111250226751603, 4.97941412507888957099089621112, 5.90633581703496709628859743985, 6.98529144610785914249417158080, 7.50528123297249961836126995042, 8.313008807830177723787394250935, 9.208110597932124863375837082776