| L(s) = 1 | + 2.63i·5-s + 0.654i·7-s + 11-s − 4.04·13-s + 5.00i·17-s + 5.62i·19-s − 3.63·23-s − 1.92·25-s + 3.36i·29-s − 8.47i·31-s − 1.72·35-s − 5.16·37-s + 1.08i·41-s + 4.31i·43-s − 1.67·47-s + ⋯ |
| L(s) = 1 | + 1.17i·5-s + 0.247i·7-s + 0.301·11-s − 1.12·13-s + 1.21i·17-s + 1.29i·19-s − 0.757·23-s − 0.385·25-s + 0.624i·29-s − 1.52i·31-s − 0.291·35-s − 0.849·37-s + 0.169i·41-s + 0.658i·43-s − 0.244·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.816 + 0.577i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.816 + 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5463154011\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5463154011\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 5 | \( 1 - 2.63iT - 5T^{2} \) |
| 7 | \( 1 - 0.654iT - 7T^{2} \) |
| 13 | \( 1 + 4.04T + 13T^{2} \) |
| 17 | \( 1 - 5.00iT - 17T^{2} \) |
| 19 | \( 1 - 5.62iT - 19T^{2} \) |
| 23 | \( 1 + 3.63T + 23T^{2} \) |
| 29 | \( 1 - 3.36iT - 29T^{2} \) |
| 31 | \( 1 + 8.47iT - 31T^{2} \) |
| 37 | \( 1 + 5.16T + 37T^{2} \) |
| 41 | \( 1 - 1.08iT - 41T^{2} \) |
| 43 | \( 1 - 4.31iT - 43T^{2} \) |
| 47 | \( 1 + 1.67T + 47T^{2} \) |
| 53 | \( 1 + 4.48iT - 53T^{2} \) |
| 59 | \( 1 + 5.40T + 59T^{2} \) |
| 61 | \( 1 + 6.90T + 61T^{2} \) |
| 67 | \( 1 + 2.94iT - 67T^{2} \) |
| 71 | \( 1 - 5.51T + 71T^{2} \) |
| 73 | \( 1 + 16.0T + 73T^{2} \) |
| 79 | \( 1 + 5.00iT - 79T^{2} \) |
| 83 | \( 1 - 16.2T + 83T^{2} \) |
| 89 | \( 1 + 9.74iT - 89T^{2} \) |
| 97 | \( 1 + 3.10T + 97T^{2} \) |
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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
−8.276358000711771086866840632888, −7.69663329905205907280721207381, −7.09887617613356505418305815068, −6.19468076386673013407877278115, −5.93427108140451365818295958643, −4.84941526343339529933779468731, −3.93721655658998077893461658328, −3.31637987618588875573821462816, −2.37694704004913219620065251342, −1.66359529838081129249659385687,
0.14249215509874973828768763189, 1.06469975140297297889510907287, 2.19424943381924472907513051966, 3.06071080298146058381125699030, 4.15085100761321666204054503938, 4.90256979414757455477442229603, 5.12943885873949589746070545876, 6.18769575701833879675019292682, 7.15813304719543831518155039303, 7.41660325777931809566656951657
