L(s) = 1 | + (−1.30 − 0.755i)2-s + (−1.57 − 0.714i)3-s + (0.141 + 0.244i)4-s + (−3.48 + 2.01i)5-s + (1.52 + 2.12i)6-s + (−0.866 − 0.5i)7-s + 2.59i·8-s + (1.98 + 2.25i)9-s + 6.07·10-s + (−0.727 − 0.420i)11-s + (−0.0481 − 0.486i)12-s + (3.38 + 1.23i)13-s + (0.755 + 1.30i)14-s + (6.93 − 0.686i)15-s + (2.24 − 3.88i)16-s − 2.30·17-s + ⋯ |
L(s) = 1 | + (−0.925 − 0.534i)2-s + (−0.911 − 0.412i)3-s + (0.0705 + 0.122i)4-s + (−1.55 + 0.899i)5-s + (0.622 + 0.868i)6-s + (−0.327 − 0.188i)7-s + 0.917i·8-s + (0.660 + 0.751i)9-s + 1.92·10-s + (−0.219 − 0.126i)11-s + (−0.0138 − 0.140i)12-s + (0.939 + 0.341i)13-s + (0.201 + 0.349i)14-s + (1.79 − 0.177i)15-s + (0.560 − 0.970i)16-s − 0.559·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.751 + 0.660i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.751 + 0.660i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0440121 - 0.116759i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0440121 - 0.116759i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (1.57 + 0.714i)T \) |
| 7 | \( 1 + (0.866 + 0.5i)T \) |
| 13 | \( 1 + (-3.38 - 1.23i)T \) |
good | 2 | \( 1 + (1.30 + 0.755i)T + (1 + 1.73i)T^{2} \) |
| 5 | \( 1 + (3.48 - 2.01i)T + (2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (0.727 + 0.420i)T + (5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + 2.30T + 17T^{2} \) |
| 19 | \( 1 - 6.65iT - 19T^{2} \) |
| 23 | \( 1 + (0.0775 + 0.134i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (1.79 - 3.10i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (6.73 - 3.88i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 6.83iT - 37T^{2} \) |
| 41 | \( 1 + (1.45 - 0.841i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (1.84 - 3.19i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-3.93 - 2.27i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + 2.68T + 53T^{2} \) |
| 59 | \( 1 + (10.1 - 5.86i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-2.56 + 4.44i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-3.17 + 1.83i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 3.96iT - 71T^{2} \) |
| 73 | \( 1 + 9.47iT - 73T^{2} \) |
| 79 | \( 1 + (2.54 - 4.40i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-10.6 - 6.12i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + 16.2iT - 89T^{2} \) |
| 97 | \( 1 + (9.08 + 5.24i)T + (48.5 + 84.0i)T^{2} \) |
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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
−10.30784131044040759457189268440, −9.089310981599971995166391122850, −8.099582587611480756418081325383, −7.51357609805137749266919258318, −6.61542038799117941375083370083, −5.65436542676913199490102386620, −4.29867728111019085397613094919, −3.30369608749556639765861627914, −1.68061854660873863636630102502, −0.15202894674600498011173042391,
0.790740999514787831888003082576, 3.54413087026872169008400489025, 4.27104316327694700515381497285, 5.22106692115784717206667593740, 6.46337907172386981640215189206, 7.26139717497153784050796823163, 8.077710735637778241362683808174, 8.896141068196698383859563986284, 9.381160133809691268730920169348, 10.57267612291482882105534676458