| L(s) = 1 | + (1.29 − 0.560i)2-s + (1.56 + 2.87i)3-s + (1.37 − 1.45i)4-s + (−0.312 − 2.21i)5-s + (3.64 + 2.85i)6-s + (0.499 + 0.667i)7-s + (0.963 − 2.65i)8-s + (−4.17 + 6.49i)9-s + (−1.64 − 2.69i)10-s + (−0.634 + 0.289i)11-s + (6.33 + 1.65i)12-s + (2.94 + 2.20i)13-s + (1.02 + 0.586i)14-s + (5.87 − 4.37i)15-s + (−0.240 − 3.99i)16-s + (1.47 − 0.105i)17-s + ⋯ |
| L(s) = 1 | + (0.918 − 0.396i)2-s + (0.905 + 1.65i)3-s + (0.685 − 0.728i)4-s + (−0.139 − 0.990i)5-s + (1.48 + 1.16i)6-s + (0.188 + 0.252i)7-s + (0.340 − 0.940i)8-s + (−1.39 + 2.16i)9-s + (−0.520 − 0.853i)10-s + (−0.191 + 0.0873i)11-s + (1.82 + 0.477i)12-s + (0.816 + 0.611i)13-s + (0.273 + 0.156i)14-s + (1.51 − 1.12i)15-s + (−0.0600 − 0.998i)16-s + (0.357 − 0.0255i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.924 - 0.381i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.924 - 0.381i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.98103 + 0.591493i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.98103 + 0.591493i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.29 + 0.560i)T \) |
| 5 | \( 1 + (0.312 + 2.21i)T \) |
| 23 | \( 1 + (-1.99 - 4.36i)T \) |
| good | 3 | \( 1 + (-1.56 - 2.87i)T + (-1.62 + 2.52i)T^{2} \) |
| 7 | \( 1 + (-0.499 - 0.667i)T + (-1.97 + 6.71i)T^{2} \) |
| 11 | \( 1 + (0.634 - 0.289i)T + (7.20 - 8.31i)T^{2} \) |
| 13 | \( 1 + (-2.94 - 2.20i)T + (3.66 + 12.4i)T^{2} \) |
| 17 | \( 1 + (-1.47 + 0.105i)T + (16.8 - 2.41i)T^{2} \) |
| 19 | \( 1 + (4.22 + 4.87i)T + (-2.70 + 18.8i)T^{2} \) |
| 29 | \( 1 + (2.22 + 1.93i)T + (4.12 + 28.7i)T^{2} \) |
| 31 | \( 1 + (1.29 + 4.41i)T + (-26.0 + 16.7i)T^{2} \) |
| 37 | \( 1 + (1.59 + 7.33i)T + (-33.6 + 15.3i)T^{2} \) |
| 41 | \( 1 + (6.82 - 4.38i)T + (17.0 - 37.2i)T^{2} \) |
| 43 | \( 1 + (-1.34 - 2.45i)T + (-23.2 + 36.1i)T^{2} \) |
| 47 | \( 1 + (2.63 - 2.63i)T - 47iT^{2} \) |
| 53 | \( 1 + (-4.10 - 5.48i)T + (-14.9 + 50.8i)T^{2} \) |
| 59 | \( 1 + (-1.62 - 11.3i)T + (-56.6 + 16.6i)T^{2} \) |
| 61 | \( 1 + (1.23 - 0.362i)T + (51.3 - 32.9i)T^{2} \) |
| 67 | \( 1 + (3.83 + 10.2i)T + (-50.6 + 43.8i)T^{2} \) |
| 71 | \( 1 + (-2.63 - 1.20i)T + (46.4 + 53.6i)T^{2} \) |
| 73 | \( 1 + (-1.40 - 0.100i)T + (72.2 + 10.3i)T^{2} \) |
| 79 | \( 1 + (1.24 + 8.64i)T + (-75.7 + 22.2i)T^{2} \) |
| 83 | \( 1 + (3.40 + 15.6i)T + (-75.4 + 34.4i)T^{2} \) |
| 89 | \( 1 + (-0.306 + 1.04i)T + (-74.8 - 48.1i)T^{2} \) |
| 97 | \( 1 + (-7.80 - 1.69i)T + (88.2 + 40.2i)T^{2} \) |
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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
−11.12935206623554427969990078069, −10.27510866821250042424693664846, −9.251002587413898932144412484367, −8.865698784340048572523206304732, −7.65715261248819654148206831341, −5.86941885506389921989199597882, −4.95046883057989544607902085163, −4.27842033042928755135272615653, −3.44762028308220395289375080527, −2.10684633318466338209630753825,
1.81296734010226266887881177695, 2.98571291580890856850496732209, 3.72498945372279762437277050739, 5.68723756734290127466445212103, 6.60385154655027953732780956960, 7.09750236360578973704346337251, 8.158739971357600475657380017854, 8.443144725332270617434984386381, 10.37954704944095985242021984609, 11.28812320833129463345898072700
