L(s) = 1 | + (−1.54 − 0.784i)3-s + (0.554 − 2.16i)5-s + (−0.608 + 0.351i)7-s + (1.77 + 2.42i)9-s + (−1.80 − 3.12i)11-s + (−1.97 − 1.14i)13-s + (−2.55 + 2.91i)15-s − 0.471i·17-s − 5.51·19-s + (1.21 − 0.0654i)21-s + (−2.68 − 1.55i)23-s + (−4.38 − 2.40i)25-s + (−0.834 − 5.12i)27-s + (0.195 + 0.339i)29-s + (−3.10 + 5.38i)31-s + ⋯ |
L(s) = 1 | + (−0.891 − 0.452i)3-s + (0.247 − 0.968i)5-s + (−0.230 + 0.132i)7-s + (0.590 + 0.807i)9-s + (−0.543 − 0.941i)11-s + (−0.549 − 0.316i)13-s + (−0.659 + 0.751i)15-s − 0.114i·17-s − 1.26·19-s + (0.265 − 0.0142i)21-s + (−0.560 − 0.323i)23-s + (−0.877 − 0.480i)25-s + (−0.160 − 0.987i)27-s + (0.0363 + 0.0630i)29-s + (−0.558 + 0.966i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.849 + 0.527i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.849 + 0.527i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.170127 - 0.597070i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.170127 - 0.597070i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.54 + 0.784i)T \) |
| 5 | \( 1 + (-0.554 + 2.16i)T \) |
good | 7 | \( 1 + (0.608 - 0.351i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (1.80 + 3.12i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (1.97 + 1.14i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + 0.471iT - 17T^{2} \) |
| 19 | \( 1 + 5.51T + 19T^{2} \) |
| 23 | \( 1 + (2.68 + 1.55i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-0.195 - 0.339i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (3.10 - 5.38i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 10.1iT - 37T^{2} \) |
| 41 | \( 1 + (-5.18 + 8.97i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-0.279 + 0.161i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-9.51 + 5.49i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 - 12.0iT - 53T^{2} \) |
| 59 | \( 1 + (-3.13 + 5.42i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-1.09 - 1.89i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-9.04 - 5.22i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 3.13T + 71T^{2} \) |
| 73 | \( 1 - 3.39iT - 73T^{2} \) |
| 79 | \( 1 + (1.66 + 2.88i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-12.2 + 7.06i)T + (41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + 4.11T + 89T^{2} \) |
| 97 | \( 1 + (-11.4 + 6.62i)T + (48.5 - 84.0i)T^{2} \) |
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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
−10.98231405062563628767648947397, −10.40235951423951785940114211213, −9.142005527528526939093523338017, −8.262384542947069140698228701578, −7.21685802411695948338446061393, −5.94650965947965258251877024214, −5.40326536911903289295194153620, −4.20413883442167683991298954921, −2.22804894497456894023499300749, −0.44891791272000737904585807679,
2.26753959095506360226976319099, 3.86723829124104199811856969152, 4.92977904785188841332038458706, 6.15969048154045505752669917945, 6.83480019829337911591000762070, 7.87290006069668464305949142715, 9.548987886200217745911439843960, 10.03066129698311449156887148070, 10.84383174108370318456115390491, 11.64143039298719002859137323458