L(s) = 1 | + (0.866 + 0.5i)3-s + (2.09 + 1.62i)7-s + (0.499 + 0.866i)9-s + (2.12 − 3.67i)11-s − 3.24i·13-s + (−3.67 − 2.12i)17-s + (−3.5 − 6.06i)19-s + (0.999 + 2.44i)21-s + (3.67 − 2.12i)23-s + 0.999i·27-s + 1.75·29-s + (4.74 − 8.21i)31-s + (3.67 − 2.12i)33-s + (2.80 − 1.62i)37-s + (1.62 − 2.80i)39-s + ⋯ |
L(s) = 1 | + (0.499 + 0.288i)3-s + (0.790 + 0.612i)7-s + (0.166 + 0.288i)9-s + (0.639 − 1.10i)11-s − 0.899i·13-s + (−0.891 − 0.514i)17-s + (−0.802 − 1.39i)19-s + (0.218 + 0.534i)21-s + (0.766 − 0.442i)23-s + 0.192i·27-s + 0.326·29-s + (0.851 − 1.47i)31-s + (0.639 − 0.369i)33-s + (0.461 − 0.266i)37-s + (0.259 − 0.449i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.788 + 0.615i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.788 + 0.615i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.251433423\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.251433423\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 + (-2.09 - 1.62i)T \) |
good | 11 | \( 1 + (-2.12 + 3.67i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + 3.24iT - 13T^{2} \) |
| 17 | \( 1 + (3.67 + 2.12i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (3.5 + 6.06i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-3.67 + 2.12i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 1.75T + 29T^{2} \) |
| 31 | \( 1 + (-4.74 + 8.21i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-2.80 + 1.62i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 4.24T + 41T^{2} \) |
| 43 | \( 1 + 3.24iT - 43T^{2} \) |
| 47 | \( 1 + (5.19 - 3i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-7.34 - 4.24i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (5.12 - 8.87i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (2.24 + 3.88i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-4.54 - 2.62i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 12.7T + 71T^{2} \) |
| 73 | \( 1 + (-8.00 - 4.62i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-5.5 - 9.52i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 10.2iT - 83T^{2} \) |
| 89 | \( 1 + (5.12 + 8.87i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 0.485iT - 97T^{2} \) |
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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
−8.705657983179153261784353616697, −8.619831245832933634290174512415, −7.62961838192802663211376129569, −6.64871406558755304009996949728, −5.83245555105698813496493370354, −4.87886069119657525836654973162, −4.21941602544691223617504858928, −2.96322776930228144192367428412, −2.35495714475570489550628458882, −0.77795509260581961541259531352,
1.47508839883264783770891687683, 1.99237552250492068938053354448, 3.48259487610441857146031951067, 4.34286884759447799192582552849, 4.88267690386088910378404687482, 6.39335599295561276844067170216, 6.81607151018801189145417291411, 7.66819345808623863928638769307, 8.428817137054532193017815688672, 9.021950019854646103344055650218