L(s) = 1 | + (1 + i)3-s − i·9-s + (3 − 3i)11-s + (−3 − 3i)13-s − 4·17-s + (−1 − i)19-s − 8i·23-s + (4 − 4i)27-s + (−3 − 3i)29-s + 6·33-s + (−3 + 3i)37-s − 6i·39-s + (−3 + 3i)43-s + 2·47-s + 7·49-s + ⋯ |
L(s) = 1 | + (0.577 + 0.577i)3-s − 0.333i·9-s + (0.904 − 0.904i)11-s + (−0.832 − 0.832i)13-s − 0.970·17-s + (−0.229 − 0.229i)19-s − 1.66i·23-s + (0.769 − 0.769i)27-s + (−0.557 − 0.557i)29-s + 1.04·33-s + (−0.493 + 0.493i)37-s − 0.960i·39-s + (−0.457 + 0.457i)43-s + 0.291·47-s + 49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.382 + 0.923i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.382 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.672370551\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.672370551\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
good | 3 | \( 1 + (-1 - i)T + 3iT^{2} \) |
| 7 | \( 1 - 7T^{2} \) |
| 11 | \( 1 + (-3 + 3i)T - 11iT^{2} \) |
| 13 | \( 1 + (3 + 3i)T + 13iT^{2} \) |
| 17 | \( 1 + 4T + 17T^{2} \) |
| 19 | \( 1 + (1 + i)T + 19iT^{2} \) |
| 23 | \( 1 + 8iT - 23T^{2} \) |
| 29 | \( 1 + (3 + 3i)T + 29iT^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + (3 - 3i)T - 37iT^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 + (3 - 3i)T - 43iT^{2} \) |
| 47 | \( 1 - 2T + 47T^{2} \) |
| 53 | \( 1 + (9 - 9i)T - 53iT^{2} \) |
| 59 | \( 1 + (-9 + 9i)T - 59iT^{2} \) |
| 61 | \( 1 + (5 + 5i)T + 61iT^{2} \) |
| 67 | \( 1 + (-3 - 3i)T + 67iT^{2} \) |
| 71 | \( 1 - 6iT - 71T^{2} \) |
| 73 | \( 1 + 6iT - 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 + (-9 - 9i)T + 83iT^{2} \) |
| 89 | \( 1 + 12iT - 89T^{2} \) |
| 97 | \( 1 - 12T + 97T^{2} \) |
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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
−9.130595015766078544735994199816, −8.669748675047618226351140687924, −7.87548490829461567352717752556, −6.69932726753547858786528710932, −6.16393250749110025375667750850, −4.92381506826448077951628032774, −4.13367813785045138142373593225, −3.25365281606540431105998969046, −2.35708580210966220561969062932, −0.58016147763625966245952406500,
1.67782821992668627847569775687, 2.21221419329512794238587704599, 3.58516211411290056779105518531, 4.50809838337175622411910204168, 5.39656596233904360094336024295, 6.69160712520191154728844575589, 7.15455050290143384610473330998, 7.82161665941878416819812405175, 8.924500959913202220969583298190, 9.297979972482170517350847655877