| L(s) = 1 | + (−0.618 − 2.40i)2-s + (−3.26 + 0.621i)3-s + (−3.66 + 2.01i)4-s + (3.51 + 7.47i)6-s + (−0.463 + 0.637i)7-s + (3.72 + 3.96i)8-s + (7.45 − 2.95i)9-s + (1.00 − 0.258i)11-s + (10.7 − 8.85i)12-s + (0.207 + 0.524i)13-s + (1.82 + 0.721i)14-s + (2.75 − 4.34i)16-s + (1.66 − 3.03i)17-s + (−11.7 − 16.1i)18-s + (−0.598 + 3.13i)19-s + ⋯ |
| L(s) = 1 | + (−0.437 − 1.70i)2-s + (−1.88 + 0.359i)3-s + (−1.83 + 1.00i)4-s + (1.43 + 3.04i)6-s + (−0.175 + 0.241i)7-s + (1.31 + 1.40i)8-s + (2.48 − 0.983i)9-s + (0.303 − 0.0778i)11-s + (3.09 − 2.55i)12-s + (0.0575 + 0.145i)13-s + (0.487 + 0.192i)14-s + (0.689 − 1.08i)16-s + (0.404 − 0.734i)17-s + (−2.76 − 3.80i)18-s + (−0.137 + 0.719i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.994 - 0.105i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.994 - 0.105i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0180867 + 0.343066i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0180867 + 0.343066i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| good | 2 | \( 1 + (0.618 + 2.40i)T + (-1.75 + 0.963i)T^{2} \) |
| 3 | \( 1 + (3.26 - 0.621i)T + (2.78 - 1.10i)T^{2} \) |
| 7 | \( 1 + (0.463 - 0.637i)T + (-2.16 - 6.65i)T^{2} \) |
| 11 | \( 1 + (-1.00 + 0.258i)T + (9.63 - 5.29i)T^{2} \) |
| 13 | \( 1 + (-0.207 - 0.524i)T + (-9.47 + 8.89i)T^{2} \) |
| 17 | \( 1 + (-1.66 + 3.03i)T + (-9.10 - 14.3i)T^{2} \) |
| 19 | \( 1 + (0.598 - 3.13i)T + (-17.6 - 6.99i)T^{2} \) |
| 23 | \( 1 + (-3.37 - 0.212i)T + (22.8 + 2.88i)T^{2} \) |
| 29 | \( 1 + (2.01 + 0.254i)T + (28.0 + 7.21i)T^{2} \) |
| 31 | \( 1 + (5.56 + 3.06i)T + (16.6 + 26.1i)T^{2} \) |
| 37 | \( 1 + (-4.67 - 2.96i)T + (15.7 + 33.4i)T^{2} \) |
| 41 | \( 1 + (0.714 + 11.3i)T + (-40.6 + 5.13i)T^{2} \) |
| 43 | \( 1 + (2.69 - 0.877i)T + (34.7 - 25.2i)T^{2} \) |
| 47 | \( 1 + (1.00 - 1.07i)T + (-2.95 - 46.9i)T^{2} \) |
| 53 | \( 1 + (2.80 + 1.31i)T + (33.7 + 40.8i)T^{2} \) |
| 59 | \( 1 + (2.27 + 2.74i)T + (-11.0 + 57.9i)T^{2} \) |
| 61 | \( 1 + (-0.470 + 7.47i)T + (-60.5 - 7.64i)T^{2} \) |
| 67 | \( 1 + (1.70 + 13.5i)T + (-64.8 + 16.6i)T^{2} \) |
| 71 | \( 1 + (-3.81 - 3.57i)T + (4.45 + 70.8i)T^{2} \) |
| 73 | \( 1 + (4.81 + 3.98i)T + (13.6 + 71.7i)T^{2} \) |
| 79 | \( 1 + (1.73 + 9.07i)T + (-73.4 + 29.0i)T^{2} \) |
| 83 | \( 1 + (-0.187 - 0.0358i)T + (77.1 + 30.5i)T^{2} \) |
| 89 | \( 1 + (-1.42 + 1.71i)T + (-16.6 - 87.4i)T^{2} \) |
| 97 | \( 1 + (-1.51 + 12.0i)T + (-93.9 - 24.1i)T^{2} \) |
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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
−10.40219120589071764370501217528, −9.679175937854577653459960067674, −9.068094044267971229677672060963, −7.52431374700720334526560515825, −6.30988155312028450282582304362, −5.32405205159115634767088393930, −4.38819425660683528491628527588, −3.41494625864808450917931483819, −1.65845266677745265235306110013, −0.36155871232299930147555821842,
1.08712711192379925039892377571, 4.23896358409201030874527833182, 5.18521678699201499400063874055, 5.82283092299428681660836959643, 6.66539660633541410706356694944, 7.12292545653687017481908128464, 8.042720386625599787997946757203, 9.238510726421426647140944716364, 10.13887853294699023001646075888, 10.93756239244513450361554005652
