| L(s) = 1 | − 2i·7-s + 3·9-s − 4·11-s + 6i·13-s − 4i·17-s − 4·19-s − 6i·23-s + 29-s − 8i·37-s − 2·41-s − 4i·43-s − 4i·47-s + 3·49-s + 2i·53-s − 8·59-s + ⋯ |
| L(s) = 1 | − 0.755i·7-s + 9-s − 1.20·11-s + 1.66i·13-s − 0.970i·17-s − 0.917·19-s − 1.25i·23-s + 0.185·29-s − 1.31i·37-s − 0.312·41-s − 0.609i·43-s − 0.583i·47-s + 0.428·49-s + 0.274i·53-s − 1.04·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.063723247\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.063723247\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 29 | \( 1 - T \) |
| good | 3 | \( 1 - 3T^{2} \) |
| 7 | \( 1 + 2iT - 7T^{2} \) |
| 11 | \( 1 + 4T + 11T^{2} \) |
| 13 | \( 1 - 6iT - 13T^{2} \) |
| 17 | \( 1 + 4iT - 17T^{2} \) |
| 19 | \( 1 + 4T + 19T^{2} \) |
| 23 | \( 1 + 6iT - 23T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 8iT - 37T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 + 4iT - 43T^{2} \) |
| 47 | \( 1 + 4iT - 47T^{2} \) |
| 53 | \( 1 - 2iT - 53T^{2} \) |
| 59 | \( 1 + 8T + 59T^{2} \) |
| 61 | \( 1 - 10T + 61T^{2} \) |
| 67 | \( 1 + 10iT - 67T^{2} \) |
| 71 | \( 1 + 8T + 71T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 - 6iT - 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 + 12iT - 97T^{2} \) |
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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
−8.559707622949417664301421051614, −7.60068513640877513632254178871, −7.02495085014998233467828999705, −6.50394676750622861292798463034, −5.29306695609008653782235124888, −4.42360334142059178851750104458, −4.04440934251699647882786590302, −2.64829727470216481005156600208, −1.78416816676095358309114768162, −0.33450060998632641076951143471,
1.34177582048891787878689287228, 2.48474427366869866890868891493, 3.29818970772638684588407964536, 4.36145640142594110115537733342, 5.29370203417765677142249418246, 5.80142436213539002183481841905, 6.72385142506222739225726006087, 7.77606296848255434911357900607, 8.074502068491007693584185571890, 8.908126706885728782043860506154
