L(s) = 1 | + (1.10 + 0.878i)2-s + (−1.07 − 1.85i)3-s + (0.455 + 1.94i)4-s + (2.26 + 1.30i)5-s + (0.443 − 2.99i)6-s + (−1.20 + 2.55i)8-s + (−0.792 + 1.37i)9-s + (1.35 + 3.43i)10-s + (3.42 − 1.97i)11-s + (3.12 − 2.92i)12-s − 1.08i·13-s − 5.59i·15-s + (−3.58 + 1.77i)16-s + (−0.274 + 0.158i)17-s + (−2.08 + 0.824i)18-s + (−2.58 + 4.47i)19-s + ⋯ |
L(s) = 1 | + (0.783 + 0.621i)2-s + (−0.618 − 1.07i)3-s + (0.227 + 0.973i)4-s + (1.01 + 0.584i)5-s + (0.181 − 1.22i)6-s + (−0.426 + 0.904i)8-s + (−0.264 + 0.457i)9-s + (0.429 + 1.08i)10-s + (1.03 − 0.596i)11-s + (0.901 − 0.845i)12-s − 0.300i·13-s − 1.44i·15-s + (−0.896 + 0.443i)16-s + (−0.0665 + 0.0384i)17-s + (−0.491 + 0.194i)18-s + (−0.593 + 1.02i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.921 - 0.387i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.921 - 0.387i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.65947 + 0.334747i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.65947 + 0.334747i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.10 - 0.878i)T \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + (1.07 + 1.85i)T + (-1.5 + 2.59i)T^{2} \) |
| 5 | \( 1 + (-2.26 - 1.30i)T + (2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-3.42 + 1.97i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + 1.08iT - 13T^{2} \) |
| 17 | \( 1 + (0.274 - 0.158i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (2.58 - 4.47i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (2.00 + 1.15i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 6.82T + 29T^{2} \) |
| 31 | \( 1 + (3.02 + 5.24i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-2 + 3.46i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 2.29iT - 41T^{2} \) |
| 43 | \( 1 - 7.23iT - 43T^{2} \) |
| 47 | \( 1 + (-2.14 + 3.70i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (5.24 + 9.08i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-5.61 - 9.72i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-4.68 - 2.70i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (2.83 - 1.63i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 + (-12.1 + 7.00i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-6.85 - 3.95i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 9.45T + 83T^{2} \) |
| 89 | \( 1 + (-5.18 - 2.99i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 9.23iT - 97T^{2} \) |
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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
−12.79467254576877559386964020840, −11.83761658569853913300812944579, −11.00077089952722258337963373378, −9.540288184713872823948188267362, −8.133375569684250340347181678668, −7.02413491947379309687627551007, −6.14390231646663746878875698868, −5.75963695715859299787951888194, −3.81900423564929487621868171937, −2.04244853998784349928015677489,
1.87318942350072451120820172682, 3.90410914203918494027730443233, 4.84483333294492159152205143447, 5.66929902527072148389353876797, 6.79131873729020584243819321415, 9.200902837458203258025997165216, 9.556871615524601340043543822164, 10.61226873241638991135691036550, 11.39207288746926622586268850329, 12.40669610065962264645701741805