| L(s) = 1 | + (0.342 + 0.939i)2-s + (−0.766 + 0.642i)4-s + (−1.76 + 0.310i)5-s + (−0.343 − 1.94i)7-s + (−0.866 − 0.500i)8-s + (−0.894 − 1.54i)10-s + (−1.99 + 3.44i)11-s + (−3.21 − 3.83i)13-s + (1.71 − 0.987i)14-s + (0.173 − 0.984i)16-s + (2.87 − 3.43i)17-s + (1.64 − 4.52i)19-s + (1.15 − 1.37i)20-s + (−3.92 − 0.691i)22-s + (−1.37 + 0.795i)23-s + ⋯ |
| L(s) = 1 | + (0.241 + 0.664i)2-s + (−0.383 + 0.321i)4-s + (−0.788 + 0.138i)5-s + (−0.129 − 0.735i)7-s + (−0.306 − 0.176i)8-s + (−0.282 − 0.489i)10-s + (−0.600 + 1.03i)11-s + (−0.892 − 1.06i)13-s + (0.457 − 0.264i)14-s + (0.0434 − 0.246i)16-s + (0.698 − 0.832i)17-s + (0.377 − 1.03i)19-s + (0.257 − 0.306i)20-s + (−0.836 − 0.147i)22-s + (−0.287 + 0.165i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0112 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0112 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.363207 - 0.367303i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.363207 - 0.367303i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.342 - 0.939i)T \) |
| 3 | \( 1 \) |
| 37 | \( 1 + (-1.03 - 5.99i)T \) |
| good | 5 | \( 1 + (1.76 - 0.310i)T + (4.69 - 1.71i)T^{2} \) |
| 7 | \( 1 + (0.343 + 1.94i)T + (-6.57 + 2.39i)T^{2} \) |
| 11 | \( 1 + (1.99 - 3.44i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (3.21 + 3.83i)T + (-2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (-2.87 + 3.43i)T + (-2.95 - 16.7i)T^{2} \) |
| 19 | \( 1 + (-1.64 + 4.52i)T + (-14.5 - 12.2i)T^{2} \) |
| 23 | \( 1 + (1.37 - 0.795i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (8.71 + 5.03i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 7.70iT - 31T^{2} \) |
| 41 | \( 1 + (0.119 - 0.100i)T + (7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 - 5.46iT - 43T^{2} \) |
| 47 | \( 1 + (5.65 + 9.78i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (0.387 - 2.20i)T + (-49.8 - 18.1i)T^{2} \) |
| 59 | \( 1 + (-0.606 - 0.106i)T + (55.4 + 20.1i)T^{2} \) |
| 61 | \( 1 + (-2.86 - 3.41i)T + (-10.5 + 60.0i)T^{2} \) |
| 67 | \( 1 + (-1.21 - 6.88i)T + (-62.9 + 22.9i)T^{2} \) |
| 71 | \( 1 + (-0.838 - 0.305i)T + (54.3 + 45.6i)T^{2} \) |
| 73 | \( 1 + 0.957T + 73T^{2} \) |
| 79 | \( 1 + (14.1 - 2.48i)T + (74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (-0.579 - 0.486i)T + (14.4 + 81.7i)T^{2} \) |
| 89 | \( 1 + (12.8 + 2.27i)T + (83.6 + 30.4i)T^{2} \) |
| 97 | \( 1 + (3.95 - 2.28i)T + (48.5 - 84.0i)T^{2} \) |
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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
−9.965498073983526953650674025560, −9.692905340444911001472652913124, −8.139453439339744610588183387107, −7.40262340414702402717941132060, −7.22572488609505994772253507237, −5.67225306553397572040741231537, −4.81712357169547775291431024564, −3.87807715429130107892766761706, −2.69192517218112614056792058775, −0.24656443554076467582363564326,
1.80880858801124309415309643698, 3.18180716806679185217093064297, 3.99969107809060826450793013777, 5.25318308985291785694111041106, 5.97741415078054684012014072874, 7.37978043055361355566681573085, 8.262631581211290466599835446255, 9.046800614424260934939818359768, 9.961945307803580196697255225808, 10.89777372571796556161761271093
