L(s) = 1 | + 3·5-s − 11-s + 4·13-s − 4·17-s − 8·23-s + 4·25-s − 7·29-s + 11·31-s − 4·37-s + 4·41-s + 2·43-s + 2·47-s − 11·53-s − 3·55-s + 7·59-s − 10·61-s + 12·65-s − 10·67-s − 6·71-s − 6·73-s + 11·79-s + 11·83-s − 12·85-s − 6·89-s + 7·97-s + 101-s + 103-s + ⋯ |
L(s) = 1 | + 1.34·5-s − 0.301·11-s + 1.10·13-s − 0.970·17-s − 1.66·23-s + 4/5·25-s − 1.29·29-s + 1.97·31-s − 0.657·37-s + 0.624·41-s + 0.304·43-s + 0.291·47-s − 1.51·53-s − 0.404·55-s + 0.911·59-s − 1.28·61-s + 1.48·65-s − 1.22·67-s − 0.712·71-s − 0.702·73-s + 1.23·79-s + 1.20·83-s − 1.30·85-s − 0.635·89-s + 0.710·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 28224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28224 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 7 T + p T^{2} \) |
| 31 | \( 1 - 11 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + 11 T + p T^{2} \) |
| 59 | \( 1 - 7 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 11 T + p T^{2} \) |
| 83 | \( 1 - 11 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 7 T + p T^{2} \) |
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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
−15.73291831064270, −14.87627170516456, −14.30223834239594, −13.69834465215811, −13.44762846971475, −13.11236309973230, −12.22544440668939, −11.83514982254222, −10.99175903198080, −10.61218956751218, −10.13232638457968, −9.329100496116177, −9.235747175970351, −8.277843312045185, −7.987234371114598, −7.057138228000399, −6.356458627424024, −6.012004355581490, −5.580176700396405, −4.675945295128332, −4.143158172915856, −3.305820030319768, −2.476859348686605, −1.917858718663878, −1.244227301367295, 0,
1.244227301367295, 1.917858718663878, 2.476859348686605, 3.305820030319768, 4.143158172915856, 4.675945295128332, 5.580176700396405, 6.012004355581490, 6.356458627424024, 7.057138228000399, 7.987234371114598, 8.277843312045185, 9.235747175970351, 9.329100496116177, 10.13232638457968, 10.61218956751218, 10.99175903198080, 11.83514982254222, 12.22544440668939, 13.11236309973230, 13.44762846971475, 13.69834465215811, 14.30223834239594, 14.87627170516456, 15.73291831064270