| L(s) = 1 | + 5-s − 2·7-s − 4·11-s − 4·17-s − 2·19-s + 2·23-s + 25-s − 8·29-s + 4·31-s − 2·35-s − 6·37-s + 10·41-s − 4·43-s − 3·49-s − 6·53-s − 4·55-s + 12·59-s − 2·61-s − 8·67-s + 8·77-s + 8·79-s + 12·83-s − 4·85-s − 10·89-s − 2·95-s + 8·97-s + 101-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.755·7-s − 1.20·11-s − 0.970·17-s − 0.458·19-s + 0.417·23-s + 1/5·25-s − 1.48·29-s + 0.718·31-s − 0.338·35-s − 0.986·37-s + 1.56·41-s − 0.609·43-s − 3/7·49-s − 0.824·53-s − 0.539·55-s + 1.56·59-s − 0.256·61-s − 0.977·67-s + 0.911·77-s + 0.900·79-s + 1.31·83-s − 0.433·85-s − 1.05·89-s − 0.205·95-s + 0.812·97-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 121680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 121680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5659148429\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5659148429\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−13.38023414265531, −13.13454976205375, −12.73181115837428, −12.27187584147888, −11.54712705798550, −10.97876486716607, −10.70770768509182, −10.15844312435451, −9.609420919192439, −9.257970388151094, −8.682577579118672, −8.163236504134094, −7.600811388155797, −7.025995519532437, −6.576564283102807, −6.021701945382270, −5.540116235960349, −4.956003127887775, −4.460481534636856, −3.701498750888730, −3.174942917377321, −2.450154644102736, −2.153097675159022, −1.245034448647423, −0.2264889395863913,
0.2264889395863913, 1.245034448647423, 2.153097675159022, 2.450154644102736, 3.174942917377321, 3.701498750888730, 4.460481534636856, 4.956003127887775, 5.540116235960349, 6.021701945382270, 6.576564283102807, 7.025995519532437, 7.600811388155797, 8.163236504134094, 8.682577579118672, 9.257970388151094, 9.609420919192439, 10.15844312435451, 10.70770768509182, 10.97876486716607, 11.54712705798550, 12.27187584147888, 12.73181115837428, 13.13454976205375, 13.38023414265531
