| L(s) = 1 | − 5·11-s − 13-s − 2·17-s − 7·19-s − 3·23-s + 6·31-s + 5·37-s + 9·41-s + 10·43-s + 13·47-s − 53-s + 4·59-s − 2·61-s + 6·67-s − 2·71-s − 4·73-s + 14·79-s + 10·83-s − 10·89-s − 8·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | − 1.50·11-s − 0.277·13-s − 0.485·17-s − 1.60·19-s − 0.625·23-s + 1.07·31-s + 0.821·37-s + 1.40·41-s + 1.52·43-s + 1.89·47-s − 0.137·53-s + 0.520·59-s − 0.256·61-s + 0.733·67-s − 0.237·71-s − 0.468·73-s + 1.57·79-s + 1.09·83-s − 1.05·89-s − 0.812·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 176400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 176400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.343314069\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.343314069\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 - 5 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 - 13 T + p T^{2} \) |
| 53 | \( 1 + T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 6 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 14 T + p T^{2} \) |
| 83 | \( 1 - 10 T + p T^{2} \) |
| 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
−13.10557364987063, −12.74082280993270, −12.31064247060389, −11.90857740935462, −11.03344820648377, −10.80367336848421, −10.52865847776598, −9.877647862432010, −9.387368525037717, −8.904032932979837, −8.225153685701360, −7.987835647065581, −7.490466191735423, −6.879810853275240, −6.317723083135512, −5.820063479780857, −5.413460983593228, −4.652957308528349, −4.269599508140101, −3.856712870272078, −2.793294558563260, −2.462441781565566, −2.164345704754763, −1.053836283786883, −0.3622213263296403,
0.3622213263296403, 1.053836283786883, 2.164345704754763, 2.462441781565566, 2.793294558563260, 3.856712870272078, 4.269599508140101, 4.652957308528349, 5.413460983593228, 5.820063479780857, 6.317723083135512, 6.879810853275240, 7.490466191735423, 7.987835647065581, 8.225153685701360, 8.904032932979837, 9.387368525037717, 9.877647862432010, 10.52865847776598, 10.80367336848421, 11.03344820648377, 11.90857740935462, 12.31064247060389, 12.74082280993270, 13.10557364987063
