L(s) = 1 | + (−1.71 + 0.246i)3-s + (−1.43 − 4.87i)7-s + (2.87 − 0.845i)9-s + (−5.44 + 4.72i)13-s + (2.19 + 0.645i)19-s + (3.65 + 7.99i)21-s + (3.27 + 3.77i)25-s + (−4.72 + 2.15i)27-s + (−0.512 − 0.444i)31-s − 11.7·37-s + (8.17 − 9.43i)39-s + (−6.94 + 10.8i)43-s + (−15.7 + 10.1i)49-s + (−3.92 − 0.564i)57-s + (−11.7 + 5.34i)61-s + ⋯ |
L(s) = 1 | + (−0.989 + 0.142i)3-s + (−0.540 − 1.84i)7-s + (0.959 − 0.281i)9-s + (−1.51 + 1.30i)13-s + (0.504 + 0.148i)19-s + (0.797 + 1.74i)21-s + (0.654 + 0.755i)25-s + (−0.909 + 0.415i)27-s + (−0.0921 − 0.0798i)31-s − 1.93·37-s + (1.30 − 1.51i)39-s + (−1.05 + 1.64i)43-s + (−2.25 + 1.44i)49-s + (−0.520 − 0.0747i)57-s + (−1.49 + 0.684i)61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.773 - 0.633i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.773 - 0.633i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0515266 + 0.144235i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0515266 + 0.144235i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.71 - 0.246i)T \) |
| 67 | \( 1 + (3.92 + 7.18i)T \) |
good | 5 | \( 1 + (-3.27 - 3.77i)T^{2} \) |
| 7 | \( 1 + (1.43 + 4.87i)T + (-5.88 + 3.78i)T^{2} \) |
| 11 | \( 1 + (-7.20 - 8.31i)T^{2} \) |
| 13 | \( 1 + (5.44 - 4.72i)T + (1.85 - 12.8i)T^{2} \) |
| 17 | \( 1 + (-7.06 - 15.4i)T^{2} \) |
| 19 | \( 1 + (-2.19 - 0.645i)T + (15.9 + 10.2i)T^{2} \) |
| 23 | \( 1 + (22.0 - 6.47i)T^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 + (0.512 + 0.444i)T + (4.41 + 30.6i)T^{2} \) |
| 37 | \( 1 + 11.7T + 37T^{2} \) |
| 41 | \( 1 + (17.0 + 37.2i)T^{2} \) |
| 43 | \( 1 + (6.94 - 10.8i)T + (-17.8 - 39.1i)T^{2} \) |
| 47 | \( 1 + (45.0 - 13.2i)T^{2} \) |
| 53 | \( 1 + (22.0 - 48.2i)T^{2} \) |
| 59 | \( 1 + (8.39 + 58.3i)T^{2} \) |
| 61 | \( 1 + (11.7 - 5.34i)T + (39.9 - 46.1i)T^{2} \) |
| 71 | \( 1 + (-29.4 + 64.5i)T^{2} \) |
| 73 | \( 1 + (-6.96 - 15.2i)T + (-47.8 + 55.1i)T^{2} \) |
| 79 | \( 1 + (-10.8 + 9.40i)T + (11.2 - 78.1i)T^{2} \) |
| 83 | \( 1 + (54.3 + 62.7i)T^{2} \) |
| 89 | \( 1 + (85.3 + 25.0i)T^{2} \) |
| 97 | \( 1 + 15.8iT - 97T^{2} \) |
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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
−10.53721849878690525814218916470, −9.902348639300758102779075002181, −9.276445965574434079569441388253, −7.60566417363487954814549924941, −7.04116902198539993806515848768, −6.48718300256981039787577813812, −5.05724864380511012999985624704, −4.42447657096343003673931237039, −3.41266020022716738275405997006, −1.44046715901648137671174425880,
0.086910742605287533092897733626, 2.15758067506660991869531148657, 3.20107933700866768980310599008, 5.07902602870690618647235757776, 5.30184795939611101480440526332, 6.31064374442271675436792945628, 7.17690358044657818746319515758, 8.231691042805558614176178193438, 9.200742395887381214729236877784, 10.01842570385802796767034496628