| L(s) = 1 | + 4·11-s − 6i·13-s + 6i·17-s + 4·19-s − 2·29-s − 8·31-s − 2i·37-s + 6·41-s − 12i·43-s − 8i·47-s + 7·49-s + 6i·53-s + 12·59-s + 14·61-s + 4i·67-s + ⋯ |
| L(s) = 1 | + 1.20·11-s − 1.66i·13-s + 1.45i·17-s + 0.917·19-s − 0.371·29-s − 1.43·31-s − 0.328i·37-s + 0.937·41-s − 1.82i·43-s − 1.16i·47-s + 49-s + 0.824i·53-s + 1.56·59-s + 1.79·61-s + 0.488i·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.854215073\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.854215073\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 7T^{2} \) |
| 11 | \( 1 - 4T + 11T^{2} \) |
| 13 | \( 1 + 6iT - 13T^{2} \) |
| 17 | \( 1 - 6iT - 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 + 8T + 31T^{2} \) |
| 37 | \( 1 + 2iT - 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 + 12iT - 43T^{2} \) |
| 47 | \( 1 + 8iT - 47T^{2} \) |
| 53 | \( 1 - 6iT - 53T^{2} \) |
| 59 | \( 1 - 12T + 59T^{2} \) |
| 61 | \( 1 - 14T + 61T^{2} \) |
| 67 | \( 1 - 4iT - 67T^{2} \) |
| 71 | \( 1 + 8T + 71T^{2} \) |
| 73 | \( 1 - 6iT - 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 + 12iT - 83T^{2} \) |
| 89 | \( 1 - 10T + 89T^{2} \) |
| 97 | \( 1 - 2iT - 97T^{2} \) |
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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
−9.094743344919859877060610482073, −8.528587581033797806845045095530, −7.59962791594882709441493335944, −6.93918352503521950128550326838, −5.77910362268210600002936997576, −5.44699920865496374295878712759, −3.96417219941928330908910833748, −3.50104030966001246223195561515, −2.12072137395380159076555044239, −0.858435137519986371005479529244,
1.12188486750357221554100410567, 2.28470657279009699079811902363, 3.52669041071811231590873734600, 4.33697446811332180746825328684, 5.20999346687128674767317548152, 6.26622151735023857247601604223, 6.98845760616785178267883674835, 7.56649741167929220289857634643, 8.796132632499125503946783245918, 9.402802984915132771648375843219
