L(s) = 1 | + (0.258 + 0.965i)2-s + (−0.866 − 0.5i)3-s + (−0.866 + 0.499i)4-s + (−1.52 − 1.52i)5-s + (0.258 − 0.965i)6-s + (2.42 + 1.05i)7-s + (−0.707 − 0.707i)8-s + (0.499 + 0.866i)9-s + (1.07 − 1.86i)10-s + (−2.31 + 0.620i)11-s + 12-s + (−0.713 − 3.53i)13-s + (−0.394 + 2.61i)14-s + (0.556 + 2.07i)15-s + (0.500 − 0.866i)16-s + (−1.44 − 2.49i)17-s + ⋯ |
L(s) = 1 | + (0.183 + 0.683i)2-s + (−0.499 − 0.288i)3-s + (−0.433 + 0.249i)4-s + (−0.680 − 0.680i)5-s + (0.105 − 0.394i)6-s + (0.916 + 0.399i)7-s + (−0.249 − 0.249i)8-s + (0.166 + 0.288i)9-s + (0.340 − 0.589i)10-s + (−0.698 + 0.187i)11-s + 0.288·12-s + (−0.197 − 0.980i)13-s + (−0.105 + 0.699i)14-s + (0.143 + 0.536i)15-s + (0.125 − 0.216i)16-s + (−0.349 − 0.605i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.509 + 0.860i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.509 + 0.860i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.799149 - 0.455544i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.799149 - 0.455544i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.258 - 0.965i)T \) |
| 3 | \( 1 + (0.866 + 0.5i)T \) |
| 7 | \( 1 + (-2.42 - 1.05i)T \) |
| 13 | \( 1 + (0.713 + 3.53i)T \) |
good | 5 | \( 1 + (1.52 + 1.52i)T + 5iT^{2} \) |
| 11 | \( 1 + (2.31 - 0.620i)T + (9.52 - 5.5i)T^{2} \) |
| 17 | \( 1 + (1.44 + 2.49i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.65 + 6.16i)T + (-16.4 - 9.5i)T^{2} \) |
| 23 | \( 1 + (-0.0337 - 0.0194i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-3.82 + 6.62i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (0.864 + 0.864i)T + 31iT^{2} \) |
| 37 | \( 1 + (-5.90 + 1.58i)T + (32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 + (4.54 - 1.21i)T + (35.5 - 20.5i)T^{2} \) |
| 43 | \( 1 + (-1.31 + 0.760i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-3.10 + 3.10i)T - 47iT^{2} \) |
| 53 | \( 1 + 11.5T + 53T^{2} \) |
| 59 | \( 1 + (1.12 + 0.301i)T + (51.0 + 29.5i)T^{2} \) |
| 61 | \( 1 + (4.42 - 2.55i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-0.334 - 1.25i)T + (-58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 + (4.09 + 1.09i)T + (61.4 + 35.5i)T^{2} \) |
| 73 | \( 1 + (9.09 - 9.09i)T - 73iT^{2} \) |
| 79 | \( 1 - 11.5T + 79T^{2} \) |
| 83 | \( 1 + (-5.24 - 5.24i)T + 83iT^{2} \) |
| 89 | \( 1 + (-0.0380 - 0.141i)T + (-77.0 + 44.5i)T^{2} \) |
| 97 | \( 1 + (-2.46 + 9.21i)T + (-84.0 - 48.5i)T^{2} \) |
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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
−10.86852984834222548610803784513, −9.632630242936556985827314782664, −8.527884134262120610371248543729, −7.907826620385399670452446196565, −7.20718287559241252663304091709, −5.91320505197102797935220577522, −4.97338940903871955520714198709, −4.51722941228767295775014390741, −2.65535222884115503257201263763, −0.55630097661133571337075968216,
1.62617261305301376383756779325, 3.25166949786545578013036502914, 4.22769512482501513609813760570, 5.07536222141382819059299345707, 6.29070858449271817985399056055, 7.44251058252364754901513047347, 8.239321813683355256092076953453, 9.415061432698055285816545415232, 10.63042226426374539143890591370, 10.75967521413953723504039862217