| L(s) = 1 | + (−2.09 − 0.803i)2-s + (2.97 − 0.155i)3-s + (2.25 + 2.02i)4-s + (−6.34 − 2.06i)6-s + (1.16 + 2.37i)7-s + (−1.05 − 2.06i)8-s + (5.82 − 0.611i)9-s + (−0.208 + 1.98i)11-s + (7.00 + 5.67i)12-s + (3.03 + 0.481i)13-s + (−0.536 − 5.91i)14-s + (−0.0912 − 0.867i)16-s + (−2.81 + 4.32i)17-s + (−12.6 − 3.39i)18-s + (−1.54 − 1.72i)19-s + ⋯ |
| L(s) = 1 | + (−1.48 − 0.568i)2-s + (1.71 − 0.0899i)3-s + (1.12 + 1.01i)4-s + (−2.59 − 0.841i)6-s + (0.441 + 0.897i)7-s + (−0.371 − 0.728i)8-s + (1.94 − 0.203i)9-s + (−0.0629 + 0.598i)11-s + (2.02 + 1.63i)12-s + (0.842 + 0.133i)13-s + (−0.143 − 1.57i)14-s + (−0.0228 − 0.216i)16-s + (−0.681 + 1.04i)17-s + (−2.98 − 0.800i)18-s + (−0.355 − 0.394i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 875 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.976 - 0.213i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 875 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.976 - 0.213i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.43679 + 0.154986i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.43679 + 0.154986i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 7 | \( 1 + (-1.16 - 2.37i)T \) |
| good | 2 | \( 1 + (2.09 + 0.803i)T + (1.48 + 1.33i)T^{2} \) |
| 3 | \( 1 + (-2.97 + 0.155i)T + (2.98 - 0.313i)T^{2} \) |
| 11 | \( 1 + (0.208 - 1.98i)T + (-10.7 - 2.28i)T^{2} \) |
| 13 | \( 1 + (-3.03 - 0.481i)T + (12.3 + 4.01i)T^{2} \) |
| 17 | \( 1 + (2.81 - 4.32i)T + (-6.91 - 15.5i)T^{2} \) |
| 19 | \( 1 + (1.54 + 1.72i)T + (-1.98 + 18.8i)T^{2} \) |
| 23 | \( 1 + (2.56 - 6.69i)T + (-17.0 - 15.3i)T^{2} \) |
| 29 | \( 1 + (-3.66 + 1.18i)T + (23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (1.15 + 5.44i)T + (-28.3 + 12.6i)T^{2} \) |
| 37 | \( 1 + (-0.346 + 0.427i)T + (-7.69 - 36.1i)T^{2} \) |
| 41 | \( 1 + (-2.97 - 4.08i)T + (-12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + (-5.39 + 5.39i)T - 43iT^{2} \) |
| 47 | \( 1 + (6.70 - 4.35i)T + (19.1 - 42.9i)T^{2} \) |
| 53 | \( 1 + (0.0200 + 0.382i)T + (-52.7 + 5.54i)T^{2} \) |
| 59 | \( 1 + (3.94 + 1.75i)T + (39.4 + 43.8i)T^{2} \) |
| 61 | \( 1 + (-0.718 - 1.61i)T + (-40.8 + 45.3i)T^{2} \) |
| 67 | \( 1 + (-0.627 - 0.407i)T + (27.2 + 61.2i)T^{2} \) |
| 71 | \( 1 + (4.40 + 13.5i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (-0.890 + 0.721i)T + (15.1 - 71.4i)T^{2} \) |
| 79 | \( 1 + (-0.465 + 2.19i)T + (-72.1 - 32.1i)T^{2} \) |
| 83 | \( 1 + (-15.1 + 7.70i)T + (48.7 - 67.1i)T^{2} \) |
| 89 | \( 1 + (7.29 - 3.24i)T + (59.5 - 66.1i)T^{2} \) |
| 97 | \( 1 + (-2.10 - 1.07i)T + (57.0 + 78.4i)T^{2} \) |
| 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
−9.766867105696672118071706346942, −9.217331224262474288175700771780, −8.634064129195935316182470193177, −8.039455325418348487061136305973, −7.44407470149667189055789120384, −6.15294409413192780041422748039, −4.43812229325818402040567756632, −3.26526339780698388920095418982, −2.17732405944558870445691662294, −1.68180385558071064602958187752,
0.994770234997208907176252892980, 2.27361423818848498259728619270, 3.54196701002438983161551990427, 4.54624023035439234254375905041, 6.35683167311856342758001004336, 7.14795269792185130290135790097, 7.905109660201524336578505326524, 8.546479144120761822757680999747, 8.879178101580817560780403949207, 9.881028525680221531963909075805
