| 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 | \( 1 + (1.72 - 1.99i)T + (-13.8 - 96.0i)T^{2} \) |
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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
−9.879498465534080433002645758153, −9.579750542787386738670151177100, −8.577289666766007388781331162816, −7.69081117490401713765216510471, −6.58489628485707211885034594603, −5.49841771525136359828992073011, −4.76090083516971705197676936917, −3.36021065783546917985965708872, −3.01040526623929205301590242349, −1.07340839019613932518630493084,
0.73183606330855512838104993902, 2.36146018987505759374825697774, 3.92604964051836177415080304487, 4.86019616537171627125546342783, 5.78524029608800167111916004818, 6.76571026245955152814828459618, 7.35040355896537341468457377284, 8.102778280650796577975215894540, 9.084194666645949424018475456991, 9.875624815225767084210249531344
