| L(s) = 1 | + (1.51 − 2.12i)3-s + (−3.83 + 0.738i)5-s + (0.145 + 2.64i)7-s + (−1.24 − 3.60i)9-s + (−4.00 + 3.15i)11-s + (−3.22 + 2.07i)13-s + (−4.23 + 9.26i)15-s + (4.61 + 4.39i)17-s + (−0.321 + 0.306i)19-s + (5.83 + 3.69i)21-s + (−2.65 − 3.99i)23-s + (9.48 − 3.79i)25-s + (−2.03 − 0.598i)27-s + (−3.71 + 1.09i)29-s + (−1.61 + 0.154i)31-s + ⋯ |
| L(s) = 1 | + (0.874 − 1.22i)3-s + (−1.71 + 0.330i)5-s + (0.0548 + 0.998i)7-s + (−0.415 − 1.20i)9-s + (−1.20 + 0.950i)11-s + (−0.895 + 0.575i)13-s + (−1.09 + 2.39i)15-s + (1.11 + 1.06i)17-s + (−0.0737 + 0.0703i)19-s + (1.27 + 0.805i)21-s + (−0.554 − 0.832i)23-s + (1.89 − 0.759i)25-s + (−0.392 − 0.115i)27-s + (−0.689 + 0.202i)29-s + (−0.290 + 0.0276i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 644 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.170 - 0.985i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 644 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.170 - 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.400728 + 0.476091i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.400728 + 0.476091i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + (-0.145 - 2.64i)T \) |
| 23 | \( 1 + (2.65 + 3.99i)T \) |
| good | 3 | \( 1 + (-1.51 + 2.12i)T + (-0.981 - 2.83i)T^{2} \) |
| 5 | \( 1 + (3.83 - 0.738i)T + (4.64 - 1.85i)T^{2} \) |
| 11 | \( 1 + (4.00 - 3.15i)T + (2.59 - 10.6i)T^{2} \) |
| 13 | \( 1 + (3.22 - 2.07i)T + (5.40 - 11.8i)T^{2} \) |
| 17 | \( 1 + (-4.61 - 4.39i)T + (0.808 + 16.9i)T^{2} \) |
| 19 | \( 1 + (0.321 - 0.306i)T + (0.904 - 18.9i)T^{2} \) |
| 29 | \( 1 + (3.71 - 1.09i)T + (24.3 - 15.6i)T^{2} \) |
| 31 | \( 1 + (1.61 - 0.154i)T + (30.4 - 5.86i)T^{2} \) |
| 37 | \( 1 + (1.36 + 3.94i)T + (-29.0 + 22.8i)T^{2} \) |
| 41 | \( 1 + (3.46 + 3.99i)T + (-5.83 + 40.5i)T^{2} \) |
| 43 | \( 1 + (2.18 + 4.78i)T + (-28.1 + 32.4i)T^{2} \) |
| 47 | \( 1 + (3.28 - 5.69i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (0.218 - 4.57i)T + (-52.7 - 5.03i)T^{2} \) |
| 59 | \( 1 + (7.58 + 3.90i)T + (34.2 + 48.0i)T^{2} \) |
| 61 | \( 1 + (-6.54 - 9.18i)T + (-19.9 + 57.6i)T^{2} \) |
| 67 | \( 1 + (-12.4 + 4.98i)T + (48.4 - 46.2i)T^{2} \) |
| 71 | \( 1 + (1.30 - 9.06i)T + (-68.1 - 20.0i)T^{2} \) |
| 73 | \( 1 + (-2.04 - 8.42i)T + (-64.8 + 33.4i)T^{2} \) |
| 79 | \( 1 + (-0.0213 - 0.448i)T + (-78.6 + 7.50i)T^{2} \) |
| 83 | \( 1 + (10.8 - 12.4i)T + (-11.8 - 82.1i)T^{2} \) |
| 89 | \( 1 + (12.3 + 1.17i)T + (87.3 + 16.8i)T^{2} \) |
| 97 | \( 1 + (-7.89 - 9.10i)T + (-13.8 + 96.0i)T^{2} \) |
| 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
−10.95197361381673862062129436376, −9.865585638668882313411790778674, −8.598764221500812653331264226950, −8.056010862839100661881510463867, −7.47723298015620306603881567888, −6.81245020151336388802003576641, −5.35171145211226214514435054862, −4.05468405271570458423401595110, −2.88040746757243011964819147573, −2.00706420196067071138754650895,
0.28933179794000392895377880272, 3.17021153725378938095218076923, 3.47055096323304888959550328487, 4.61687490324988509724200824213, 5.24281556162130564290837255237, 7.29456328371717250720265771217, 7.924532127558048353308048996984, 8.330984020008099406351031519933, 9.620992397498409799535982327404, 10.19101643042168921311547380983
