| L(s) = 1 | + 5-s − 4·7-s − 4·11-s + 4·13-s + 8·17-s − 8·19-s + 4·23-s + 25-s + 2·29-s − 4·35-s + 2·37-s + 2·41-s + 10·43-s − 8·47-s + 9·49-s + 12·53-s − 4·55-s − 4·59-s − 10·61-s + 4·65-s + 8·67-s + 71-s + 2·73-s + 16·77-s + 12·79-s − 14·83-s + 8·85-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 1.51·7-s − 1.20·11-s + 1.10·13-s + 1.94·17-s − 1.83·19-s + 0.834·23-s + 1/5·25-s + 0.371·29-s − 0.676·35-s + 0.328·37-s + 0.312·41-s + 1.52·43-s − 1.16·47-s + 9/7·49-s + 1.64·53-s − 0.539·55-s − 0.520·59-s − 1.28·61-s + 0.496·65-s + 0.977·67-s + 0.118·71-s + 0.234·73-s + 1.82·77-s + 1.35·79-s − 1.53·83-s + 0.867·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 204480 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 204480 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.305379294\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.305379294\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 71 | \( 1 - T \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 8 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 + 14 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 - 12 T + p T^{2} \) |
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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
−12.90716816855898, −12.75014106025946, −12.34883141957344, −11.69839927831405, −10.85171828362281, −10.75146353994945, −10.12901619649199, −9.950562384815259, −9.213107190566540, −8.912133727194408, −8.297921240465040, −7.807135437928438, −7.334974521321830, −6.607385643380337, −6.274277171046293, −5.850657651717840, −5.413424790707403, −4.770811404251641, −4.055200948868430, −3.508266288372952, −3.039463889314647, −2.581060058421556, −1.913560288763252, −0.9903781338173756, −0.4976096359019840,
0.4976096359019840, 0.9903781338173756, 1.913560288763252, 2.581060058421556, 3.039463889314647, 3.508266288372952, 4.055200948868430, 4.770811404251641, 5.413424790707403, 5.850657651717840, 6.274277171046293, 6.607385643380337, 7.334974521321830, 7.807135437928438, 8.297921240465040, 8.912133727194408, 9.213107190566540, 9.950562384815259, 10.12901619649199, 10.75146353994945, 10.85171828362281, 11.69839927831405, 12.34883141957344, 12.75014106025946, 12.90716816855898
