| L(s) = 1 | − 2·7-s − 2·11-s + 13-s + 2·17-s − 2·19-s + 5·23-s + 5·29-s − 2·37-s − 2·41-s − 43-s + 6·47-s − 3·49-s + 3·53-s + 2·59-s − 3·61-s − 2·67-s + 8·71-s + 10·73-s + 4·77-s + 7·79-s + 2·83-s + 8·89-s − 2·91-s − 18·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | − 0.755·7-s − 0.603·11-s + 0.277·13-s + 0.485·17-s − 0.458·19-s + 1.04·23-s + 0.928·29-s − 0.328·37-s − 0.312·41-s − 0.152·43-s + 0.875·47-s − 3/7·49-s + 0.412·53-s + 0.260·59-s − 0.384·61-s − 0.244·67-s + 0.949·71-s + 1.17·73-s + 0.455·77-s + 0.787·79-s + 0.219·83-s + 0.847·89-s − 0.209·91-s − 1.82·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 187200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 187200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.106653148\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.106653148\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 5 T + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 - 2 T + p T^{2} \) |
| 61 | \( 1 + 3 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 7 T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 - 8 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−13.12265234125226, −12.72256877174048, −12.15365716775393, −11.91261068700119, −11.07578047356102, −10.76692879280949, −10.35963745339668, −9.769393088690339, −9.432055201930676, −8.801375313869948, −8.392576990893781, −7.905084935539063, −7.278860790404306, −6.850840437393111, −6.329473710365096, −5.912571915602076, −5.164494542440817, −4.923225158900286, −4.147876774151554, −3.549478954653587, −3.099276746240543, −2.548008768385313, −1.913227478441396, −1.036307738660741, −0.4699114272233717,
0.4699114272233717, 1.036307738660741, 1.913227478441396, 2.548008768385313, 3.099276746240543, 3.549478954653587, 4.147876774151554, 4.923225158900286, 5.164494542440817, 5.912571915602076, 6.329473710365096, 6.850840437393111, 7.278860790404306, 7.905084935539063, 8.392576990893781, 8.801375313869948, 9.432055201930676, 9.769393088690339, 10.35963745339668, 10.76692879280949, 11.07578047356102, 11.91261068700119, 12.15365716775393, 12.72256877174048, 13.12265234125226
