L(s) = 1 | + (−1.88 − 1.88i)2-s + (2.65 − 1.39i)3-s + 3.12i·4-s + (−4.96 + 0.551i)5-s + (−7.64 − 2.36i)6-s + (−6.68 − 2.09i)7-s + (−1.65 + 1.65i)8-s + (5.08 − 7.42i)9-s + (10.4 + 8.33i)10-s − 17.9i·11-s + (4.36 + 8.28i)12-s + (−11.1 + 11.1i)13-s + (8.66 + 16.5i)14-s + (−12.4 + 8.41i)15-s + 18.7·16-s + (0.666 − 0.666i)17-s + ⋯ |
L(s) = 1 | + (−0.943 − 0.943i)2-s + (0.884 − 0.466i)3-s + 0.780i·4-s + (−0.993 + 0.110i)5-s + (−1.27 − 0.394i)6-s + (−0.954 − 0.298i)7-s + (−0.207 + 0.207i)8-s + (0.564 − 0.825i)9-s + (1.04 + 0.833i)10-s − 1.62i·11-s + (0.363 + 0.690i)12-s + (−0.856 + 0.856i)13-s + (0.618 + 1.18i)14-s + (−0.827 + 0.561i)15-s + 1.17·16-s + (0.0392 − 0.0392i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.966 - 0.258i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.966 - 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.0719889 + 0.548218i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0719889 + 0.548218i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-2.65 + 1.39i)T \) |
| 5 | \( 1 + (4.96 - 0.551i)T \) |
| 7 | \( 1 + (6.68 + 2.09i)T \) |
good | 2 | \( 1 + (1.88 + 1.88i)T + 4iT^{2} \) |
| 11 | \( 1 + 17.9iT - 121T^{2} \) |
| 13 | \( 1 + (11.1 - 11.1i)T - 169iT^{2} \) |
| 17 | \( 1 + (-0.666 + 0.666i)T - 289iT^{2} \) |
| 19 | \( 1 + 10.8T + 361T^{2} \) |
| 23 | \( 1 + (-2.20 + 2.20i)T - 529iT^{2} \) |
| 29 | \( 1 - 22.9T + 841T^{2} \) |
| 31 | \( 1 + 26.1iT - 961T^{2} \) |
| 37 | \( 1 + (-41.6 + 41.6i)T - 1.36e3iT^{2} \) |
| 41 | \( 1 + 6.85T + 1.68e3T^{2} \) |
| 43 | \( 1 + (37.6 + 37.6i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 + (5.55 - 5.55i)T - 2.20e3iT^{2} \) |
| 53 | \( 1 + (32.4 - 32.4i)T - 2.80e3iT^{2} \) |
| 59 | \( 1 + 99.8iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 44.6iT - 3.72e3T^{2} \) |
| 67 | \( 1 + (18.0 - 18.0i)T - 4.48e3iT^{2} \) |
| 71 | \( 1 - 6.35iT - 5.04e3T^{2} \) |
| 73 | \( 1 + (-55.3 + 55.3i)T - 5.32e3iT^{2} \) |
| 79 | \( 1 - 59.7iT - 6.24e3T^{2} \) |
| 83 | \( 1 + (-42.2 - 42.2i)T + 6.88e3iT^{2} \) |
| 89 | \( 1 - 58.4iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (-11.1 - 11.1i)T + 9.40e3iT^{2} \) |
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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
−12.69275705818480590664303527106, −11.79314431082603610125109112982, −10.79838220309669162742965487623, −9.587020572938952075485502913045, −8.741029273575659819508368526885, −7.81118890557277240348802019547, −6.49002560391525953886740861652, −3.77142129336238108500239155323, −2.67425565643853017459702961540, −0.49118799839339151465359996626,
3.08203101237643442036237517448, 4.69412310814543018012443475035, 6.78491862404626686857245743535, 7.65086336052459160547815663407, 8.511205568525934425463365066978, 9.625358118826273914340350245083, 10.21984420330216823063395698564, 12.25151505767957848199453467847, 12.93265711248860139626084823460, 14.89240097877874507298909716733