| L(s) = 1 | + (1.05 + 0.358i)3-s + (−0.174 + 2.65i)5-s + (0.509 + 2.59i)7-s + (−1.39 − 1.06i)9-s + (2.60 + 2.28i)11-s + (−2.91 − 2.91i)13-s + (−1.13 + 2.74i)15-s + (0.0622 + 4.12i)17-s + (−3.08 + 0.406i)19-s + (−0.392 + 2.92i)21-s + (−0.287 − 0.847i)23-s + (−2.07 − 0.273i)25-s + (−2.94 − 4.40i)27-s + (3.50 + 2.33i)29-s + (−0.542 + 1.59i)31-s + ⋯ |
| L(s) = 1 | + (0.609 + 0.206i)3-s + (−0.0779 + 1.18i)5-s + (0.192 + 0.981i)7-s + (−0.464 − 0.356i)9-s + (0.785 + 0.689i)11-s + (−0.808 − 0.808i)13-s + (−0.293 + 0.708i)15-s + (0.0150 + 0.999i)17-s + (−0.708 + 0.0932i)19-s + (−0.0856 + 0.637i)21-s + (−0.0599 − 0.176i)23-s + (−0.415 − 0.0547i)25-s + (−0.566 − 0.848i)27-s + (0.650 + 0.434i)29-s + (−0.0975 + 0.287i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 476 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0184 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 476 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0184 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.09909 + 1.11957i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.09909 + 1.11957i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + (-0.509 - 2.59i)T \) |
| 17 | \( 1 + (-0.0622 - 4.12i)T \) |
| good | 3 | \( 1 + (-1.05 - 0.358i)T + (2.38 + 1.82i)T^{2} \) |
| 5 | \( 1 + (0.174 - 2.65i)T + (-4.95 - 0.652i)T^{2} \) |
| 11 | \( 1 + (-2.60 - 2.28i)T + (1.43 + 10.9i)T^{2} \) |
| 13 | \( 1 + (2.91 + 2.91i)T + 13iT^{2} \) |
| 19 | \( 1 + (3.08 - 0.406i)T + (18.3 - 4.91i)T^{2} \) |
| 23 | \( 1 + (0.287 + 0.847i)T + (-18.2 + 14.0i)T^{2} \) |
| 29 | \( 1 + (-3.50 - 2.33i)T + (11.0 + 26.7i)T^{2} \) |
| 31 | \( 1 + (0.542 - 1.59i)T + (-24.5 - 18.8i)T^{2} \) |
| 37 | \( 1 + (-6.08 - 6.93i)T + (-4.82 + 36.6i)T^{2} \) |
| 41 | \( 1 + (-10.4 + 6.96i)T + (15.6 - 37.8i)T^{2} \) |
| 43 | \( 1 + (-10.3 + 4.28i)T + (30.4 - 30.4i)T^{2} \) |
| 47 | \( 1 + (-0.353 + 1.31i)T + (-40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (3.32 - 2.55i)T + (13.7 - 51.1i)T^{2} \) |
| 59 | \( 1 + (-1.56 + 11.8i)T + (-56.9 - 15.2i)T^{2} \) |
| 61 | \( 1 + (4.20 - 8.52i)T + (-37.1 - 48.3i)T^{2} \) |
| 67 | \( 1 + (1.01 + 0.586i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-9.27 + 1.84i)T + (65.5 - 27.1i)T^{2} \) |
| 73 | \( 1 + (4.58 - 2.26i)T + (44.4 - 57.9i)T^{2} \) |
| 79 | \( 1 + (-5.27 + 1.79i)T + (62.6 - 48.0i)T^{2} \) |
| 83 | \( 1 + (11.9 + 4.93i)T + (58.6 + 58.6i)T^{2} \) |
| 89 | \( 1 + (-4.39 - 1.17i)T + (77.0 + 44.5i)T^{2} \) |
| 97 | \( 1 + (-2.24 + 3.36i)T + (-37.1 - 89.6i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.12870500756442518168639204344, −10.31950270680372385057105976075, −9.407879680647608595506248238290, −8.593029179650572394873139684098, −7.67903943058932653391759776645, −6.57696104634388092772606120560, −5.78072904460322000942985255554, −4.28750405409406037553720007887, −3.08122984474010748634679821332, −2.28200307915975909766954828976,
0.929285739125622921034301028888, 2.51186539892674509204735494526, 4.08908406520177363677478787140, 4.78799769037562240564718552361, 6.12185898339864129333584417753, 7.38446301746694314902285984770, 8.061189600113841270704166388598, 9.073614295990418588636947151447, 9.493735636087042491455633500575, 10.96089955959644022010799407147
