| L(s) = 1 | + (−0.142 + 0.989i)2-s + (−0.415 − 0.909i)3-s + (−0.959 − 0.281i)4-s + (−0.463 − 0.535i)5-s + (0.959 − 0.281i)6-s + (−0.841 − 0.540i)7-s + (0.415 − 0.909i)8-s + (−0.654 + 0.755i)9-s + (0.595 − 0.382i)10-s + (0.451 + 3.14i)11-s + (0.142 + 0.989i)12-s + (−1.29 + 0.831i)13-s + (0.654 − 0.755i)14-s + (−0.294 + 0.644i)15-s + (0.841 + 0.540i)16-s + (−0.0911 + 0.0267i)17-s + ⋯ |
| L(s) = 1 | + (−0.100 + 0.699i)2-s + (−0.239 − 0.525i)3-s + (−0.479 − 0.140i)4-s + (−0.207 − 0.239i)5-s + (0.391 − 0.115i)6-s + (−0.317 − 0.204i)7-s + (0.146 − 0.321i)8-s + (−0.218 + 0.251i)9-s + (0.188 − 0.121i)10-s + (0.136 + 0.947i)11-s + (0.0410 + 0.285i)12-s + (−0.358 + 0.230i)13-s + (0.175 − 0.201i)14-s + (−0.0759 + 0.166i)15-s + (0.210 + 0.135i)16-s + (−0.0221 + 0.00648i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.969 + 0.243i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.969 + 0.243i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.13368 - 0.140064i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.13368 - 0.140064i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.142 - 0.989i)T \) |
| 3 | \( 1 + (0.415 + 0.909i)T \) |
| 7 | \( 1 + (0.841 + 0.540i)T \) |
| 23 | \( 1 + (-0.668 + 4.74i)T \) |
| good | 5 | \( 1 + (0.463 + 0.535i)T + (-0.711 + 4.94i)T^{2} \) |
| 11 | \( 1 + (-0.451 - 3.14i)T + (-10.5 + 3.09i)T^{2} \) |
| 13 | \( 1 + (1.29 - 0.831i)T + (5.40 - 11.8i)T^{2} \) |
| 17 | \( 1 + (0.0911 - 0.0267i)T + (14.3 - 9.19i)T^{2} \) |
| 19 | \( 1 + (-7.28 - 2.13i)T + (15.9 + 10.2i)T^{2} \) |
| 29 | \( 1 + (-3.30 + 0.970i)T + (24.3 - 15.6i)T^{2} \) |
| 31 | \( 1 + (-1.18 + 2.58i)T + (-20.3 - 23.4i)T^{2} \) |
| 37 | \( 1 + (-7.66 + 8.85i)T + (-5.26 - 36.6i)T^{2} \) |
| 41 | \( 1 + (2.14 + 2.47i)T + (-5.83 + 40.5i)T^{2} \) |
| 43 | \( 1 + (5.39 + 11.8i)T + (-28.1 + 32.4i)T^{2} \) |
| 47 | \( 1 - 0.0496T + 47T^{2} \) |
| 53 | \( 1 + (-7.41 - 4.76i)T + (22.0 + 48.2i)T^{2} \) |
| 59 | \( 1 + (-4.45 + 2.86i)T + (24.5 - 53.6i)T^{2} \) |
| 61 | \( 1 + (3.10 - 6.79i)T + (-39.9 - 46.1i)T^{2} \) |
| 67 | \( 1 + (0.391 - 2.72i)T + (-64.2 - 18.8i)T^{2} \) |
| 71 | \( 1 + (-0.224 + 1.56i)T + (-68.1 - 20.0i)T^{2} \) |
| 73 | \( 1 + (-2.70 - 0.793i)T + (61.4 + 39.4i)T^{2} \) |
| 79 | \( 1 + (-7.77 + 4.99i)T + (32.8 - 71.8i)T^{2} \) |
| 83 | \( 1 + (-2.82 + 3.26i)T + (-11.8 - 82.1i)T^{2} \) |
| 89 | \( 1 + (-7.43 - 16.2i)T + (-58.2 + 67.2i)T^{2} \) |
| 97 | \( 1 + (1.72 + 1.99i)T + (-13.8 + 96.0i)T^{2} \) |
| 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
−9.875624815225767084210249531344, −9.084194666645949424018475456991, −8.102778280650796577975215894540, −7.35040355896537341468457377284, −6.76571026245955152814828459618, −5.78524029608800167111916004818, −4.86019616537171627125546342783, −3.92604964051836177415080304487, −2.36146018987505759374825697774, −0.73183606330855512838104993902,
1.07340839019613932518630493084, 3.01040526623929205301590242349, 3.36021065783546917985965708872, 4.76090083516971705197676936917, 5.49841771525136359828992073011, 6.58489628485707211885034594603, 7.69081117490401713765216510471, 8.577289666766007388781331162816, 9.579750542787386738670151177100, 9.879498465534080433002645758153
