L(s) = 1 | + (−0.866 + 0.5i)2-s + (0.499 − 0.866i)4-s + (−1.80 − 3.13i)5-s + (2.14 + 1.54i)7-s + 0.999i·8-s + (3.13 + 1.80i)10-s + (1.73 + 1.00i)11-s + 3.40i·13-s + (−2.63 − 0.266i)14-s + (−0.5 − 0.866i)16-s + (−3.08 + 5.34i)17-s + (0.877 − 0.506i)19-s − 3.61·20-s − 2.00·22-s + (2.62 − 1.51i)23-s + ⋯ |
L(s) = 1 | + (−0.612 + 0.353i)2-s + (0.249 − 0.433i)4-s + (−0.809 − 1.40i)5-s + (0.811 + 0.584i)7-s + 0.353i·8-s + (0.991 + 0.572i)10-s + (0.523 + 0.302i)11-s + 0.945i·13-s + (−0.703 − 0.0713i)14-s + (−0.125 − 0.216i)16-s + (−0.748 + 1.29i)17-s + (0.201 − 0.116i)19-s − 0.809·20-s − 0.427·22-s + (0.546 − 0.315i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.852 - 0.522i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.852 - 0.522i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.093104641\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.093104641\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.866 - 0.5i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-2.14 - 1.54i)T \) |
good | 5 | \( 1 + (1.80 + 3.13i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-1.73 - 1.00i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 - 3.40iT - 13T^{2} \) |
| 17 | \( 1 + (3.08 - 5.34i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.877 + 0.506i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.62 + 1.51i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 5.82iT - 29T^{2} \) |
| 31 | \( 1 + (0.787 + 0.454i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-3.66 - 6.35i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 5.70T + 41T^{2} \) |
| 43 | \( 1 - 4.79T + 43T^{2} \) |
| 47 | \( 1 + (-1.11 - 1.93i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-7.58 - 4.37i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (4.49 - 7.78i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-12.7 + 7.35i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-4.15 + 7.20i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 0.466iT - 71T^{2} \) |
| 73 | \( 1 + (3.65 + 2.10i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (1.91 + 3.31i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 8.00T + 83T^{2} \) |
| 89 | \( 1 + (2.39 + 4.14i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 11.7iT - 97T^{2} \) |
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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
−9.430122545573737980023521555609, −8.988681331495182881231219979817, −8.307964098755480377919945652630, −7.74511627036207283786564131731, −6.62479125147864642296195136839, −5.67299685620496983237590875840, −4.58479148139960754842649769243, −4.16002649874358824930381222615, −2.13607618781245816917332079682, −1.06979286246290929638337906263,
0.75310422547426109593166262494, 2.45836994246500995586101616295, 3.35299410018502529063826514157, 4.21628822429571350842556305124, 5.51627440903124655559016066650, 6.96688830954652411317709675118, 7.19882836409483883312997929206, 8.024901824205572998524163233324, 8.900469235804347949934106444921, 9.899304727554841556015453776485