L(s) = 1 | + (−0.866 − 0.5i)2-s + (−0.5 + 0.866i)3-s + (0.499 + 0.866i)4-s + 2i·5-s + (0.866 − 0.499i)6-s + (−1.73 + i)7-s − 0.999i·8-s + (−0.499 − 0.866i)9-s + (1 − 1.73i)10-s − 0.999·12-s + 1.99·14-s + (−1.73 − i)15-s + (−0.5 + 0.866i)16-s + (1 + 1.73i)17-s + 0.999i·18-s + (−5.19 + 3i)19-s + ⋯ |
L(s) = 1 | + (−0.612 − 0.353i)2-s + (−0.288 + 0.499i)3-s + (0.249 + 0.433i)4-s + 0.894i·5-s + (0.353 − 0.204i)6-s + (−0.654 + 0.377i)7-s − 0.353i·8-s + (−0.166 − 0.288i)9-s + (0.316 − 0.547i)10-s − 0.288·12-s + 0.534·14-s + (−0.447 − 0.258i)15-s + (−0.125 + 0.216i)16-s + (0.242 + 0.420i)17-s + 0.235i·18-s + (−1.19 + 0.688i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.967 + 0.252i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.967 + 0.252i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.2601932850\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2601932850\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (0.5 - 0.866i)T \) |
| 13 | \( 1 \) |
good | 5 | \( 1 - 2iT - 5T^{2} \) |
| 7 | \( 1 + (1.73 - i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + (-1 - 1.73i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (5.19 - 3i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (2 - 3.46i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-5 + 8.66i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 - 10iT - 31T^{2} \) |
| 37 | \( 1 + (6.92 + 4i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (8.66 + 5i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (2 + 3.46i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + 12iT - 47T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 + (-3.46 + 2i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (1 + 1.73i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-1.73 - i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + 4iT - 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 4iT - 83T^{2} \) |
| 89 | \( 1 + (-5.19 - 3i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (10.3 - 6i)T + (48.5 - 84.0i)T^{2} \) |
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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
−10.37946866545874104232843745339, −9.845353774856940735175559953120, −8.817998564185614904695178291591, −8.149719483300746943404224830070, −6.89035366029564923123428802745, −6.38540297357788949139437156721, −5.32117634263532148084616147780, −3.89049741399223370027194030881, −3.17735851809504378545212035113, −1.98203221996889590288056288733,
0.15234640115352282176719972337, 1.38927280885199105712802525945, 2.85088454661961585487477555243, 4.42880489427841828127163397409, 5.24153708373649841930163576622, 6.44696323468050826457177793133, 6.81152903187538649792713412488, 7.987029396650742163451092542228, 8.584021761114167999738152595145, 9.400773274354392489045690357083