L(s) = 1 | − 1.41·2-s + (−1.41 − i)3-s + 2.00·4-s + (2.00 + 1.41i)6-s − 2.82·8-s + (1.00 + 2.82i)9-s − 2.82i·11-s + (−2.82 − 2.00i)12-s + 4.00·16-s + 5.65·17-s + (−1.41 − 4.00i)18-s − 2·19-s + 4.00i·22-s + (4.00 + 2.82i)24-s + (1.41 − 5.00i)27-s + ⋯ |
L(s) = 1 | − 1.00·2-s + (−0.816 − 0.577i)3-s + 1.00·4-s + (0.816 + 0.577i)6-s − 1.00·8-s + (0.333 + 0.942i)9-s − 0.852i·11-s + (−0.816 − 0.577i)12-s + 1.00·16-s + 1.37·17-s + (−0.333 − 0.942i)18-s − 0.458·19-s + 0.852i·22-s + (0.816 + 0.577i)24-s + (0.272 − 0.962i)27-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.151 + 0.988i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.151 + 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.388438 - 0.452393i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.388438 - 0.452393i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + 1.41T \) |
| 3 | \( 1 + (1.41 + i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 + 2.82iT - 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 - 5.65T + 17T^{2} \) |
| 19 | \( 1 + 2T + 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 + 11.3iT - 41T^{2} \) |
| 43 | \( 1 + 10iT - 43T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 + 14.1iT - 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 + 14iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 2iT - 73T^{2} \) |
| 79 | \( 1 - 79T^{2} \) |
| 83 | \( 1 - 2.82T + 83T^{2} \) |
| 89 | \( 1 + 5.65iT - 89T^{2} \) |
| 97 | \( 1 - 10iT - 97T^{2} \) |
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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
−10.55876245267968492484304726491, −9.646095475554709009324864519247, −8.528426301559608627421176067605, −7.81140000509226728069195917769, −6.93750669264080031930402042396, −6.04937660383398479623682106568, −5.28014977579128798028461096682, −3.44425681039095049905584847259, −1.94441140740519819610709060421, −0.56376035177458791306594881388,
1.30621410429009605960937164765, 3.03707936129206853863230936668, 4.42399386732338580105600213075, 5.60219644515789268466755375380, 6.46808119216902061846200003390, 7.39691577676555378091238245052, 8.327874802076235216812068774716, 9.534559497696618641549018501835, 9.906252985558840285303062381073, 10.72841503640906665930255540795