| L(s) = 1 | + 3.23i·3-s + i·7-s − 7.47·9-s + 0.236·11-s + 1.23i·13-s − 2.47i·17-s − 4.47·19-s − 3.23·21-s − 6.23i·23-s − 14.4i·27-s − 5·29-s − 3.70·31-s + 0.763i·33-s − 3i·37-s − 4.00·39-s + ⋯ |
| L(s) = 1 | + 1.86i·3-s + 0.377i·7-s − 2.49·9-s + 0.0711·11-s + 0.342i·13-s − 0.599i·17-s − 1.02·19-s − 0.706·21-s − 1.30i·23-s − 2.78i·27-s − 0.928·29-s − 0.666·31-s + 0.132i·33-s − 0.493i·37-s − 0.640·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1397718465\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1397718465\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - iT \) |
| good | 3 | \( 1 - 3.23iT - 3T^{2} \) |
| 11 | \( 1 - 0.236T + 11T^{2} \) |
| 13 | \( 1 - 1.23iT - 13T^{2} \) |
| 17 | \( 1 + 2.47iT - 17T^{2} \) |
| 19 | \( 1 + 4.47T + 19T^{2} \) |
| 23 | \( 1 + 6.23iT - 23T^{2} \) |
| 29 | \( 1 + 5T + 29T^{2} \) |
| 31 | \( 1 + 3.70T + 31T^{2} \) |
| 37 | \( 1 + 3iT - 37T^{2} \) |
| 41 | \( 1 - 4.76T + 41T^{2} \) |
| 43 | \( 1 + 1.76iT - 43T^{2} \) |
| 47 | \( 1 + 2iT - 47T^{2} \) |
| 53 | \( 1 - 8.47iT - 53T^{2} \) |
| 59 | \( 1 - 11.7T + 59T^{2} \) |
| 61 | \( 1 + 9.70T + 61T^{2} \) |
| 67 | \( 1 + 4.23iT - 67T^{2} \) |
| 71 | \( 1 + 8.70T + 71T^{2} \) |
| 73 | \( 1 + 8.76iT - 73T^{2} \) |
| 79 | \( 1 + 11.1T + 79T^{2} \) |
| 83 | \( 1 - 7.70iT - 83T^{2} \) |
| 89 | \( 1 + 17.2T + 89T^{2} \) |
| 97 | \( 1 + 5.23iT - 97T^{2} \) |
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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
−8.921952498810517910930307344427, −8.283541047884558335481791244563, −7.13362542137320029036728186546, −6.05616830021155433746266937816, −5.47942157351040370949303088165, −4.52485772829227384204003152822, −4.13144706755480360444829486602, −3.11771625041123512885857908301, −2.24949752254809086456957177626, −0.04413071265307839223630961030,
1.27178433283752652936136872820, 2.00832736100600395018865565229, 3.04953882838238640811421960804, 4.06221795813139764516577667612, 5.45920589556857131676084899165, 5.98479716112337461782433213735, 6.82876511315749359168968123401, 7.36389882689012113726433269499, 8.036929703147324235629660299509, 8.616607872958680640775885536164
