L(s) = 1 | + (−0.866 + 0.5i)3-s + (−0.133 + 2.23i)5-s + (−1.73 + 2i)7-s + (0.499 − 0.866i)9-s + (−2.5 − 4.33i)11-s + i·13-s + (−1 − 1.99i)15-s + (1.73 − i)17-s + (−3.5 + 6.06i)19-s + (0.499 − 2.59i)21-s + (2.59 + 1.5i)23-s + (−4.96 − 0.598i)25-s + 0.999i·27-s + (−3 − 5.19i)31-s + (4.33 + 2.5i)33-s + ⋯ |
L(s) = 1 | + (−0.499 + 0.288i)3-s + (−0.0599 + 0.998i)5-s + (−0.654 + 0.755i)7-s + (0.166 − 0.288i)9-s + (−0.753 − 1.30i)11-s + 0.277i·13-s + (−0.258 − 0.516i)15-s + (0.420 − 0.242i)17-s + (−0.802 + 1.39i)19-s + (0.109 − 0.566i)21-s + (0.541 + 0.312i)23-s + (−0.992 − 0.119i)25-s + 0.192i·27-s + (−0.538 − 0.933i)31-s + (0.753 + 0.435i)33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.185 + 0.982i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.185 + 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.1991560272\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1991560272\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.866 - 0.5i)T \) |
| 5 | \( 1 + (0.133 - 2.23i)T \) |
| 7 | \( 1 + (1.73 - 2i)T \) |
good | 11 | \( 1 + (2.5 + 4.33i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 - iT - 13T^{2} \) |
| 17 | \( 1 + (-1.73 + i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (3.5 - 6.06i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.59 - 1.5i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + (3 + 5.19i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (4.33 + 2.5i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 9T + 41T^{2} \) |
| 43 | \( 1 + 10iT - 43T^{2} \) |
| 47 | \( 1 + (-11.2 - 6.5i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (0.866 - 0.5i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (2 + 3.46i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-1 + 1.73i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-5.19 + 3i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 2T + 71T^{2} \) |
| 73 | \( 1 + (-3.46 + 2i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-7 + 12.1i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 10iT - 83T^{2} \) |
| 89 | \( 1 + (-5 + 8.66i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 8iT - 97T^{2} \) |
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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
−9.200449053453158674177060679234, −8.371696637461879765410695179805, −7.48552468034025788283708037369, −6.53434189670913029978358451985, −5.85670671530258892519594174469, −5.35820863686530138571181836972, −3.81447691735386871849139050951, −3.23041170943549989491939807696, −2.14484910464069273828233967262, −0.086552679838786349602083131944,
1.17065167745426718362236006538, 2.48153535595883915154938075427, 3.84958977739073368868564389872, 4.82862322259163020898967750542, 5.24999239924552493545243682564, 6.54384092485111405739020370627, 7.09987391035278547406145804921, 7.907339467386117203086677486550, 8.797404749993547864046682831314, 9.625257672016212720857581680350