| L(s) = 1 | + 4·11-s + 6·17-s − 6·19-s − 23-s − 5·25-s − 10·29-s + 4·31-s − 2·37-s − 10·41-s + 4·43-s − 12·47-s + 6·53-s + 2·59-s − 8·71-s + 6·73-s + 8·79-s + 14·83-s − 14·89-s + 2·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | + 1.20·11-s + 1.45·17-s − 1.37·19-s − 0.208·23-s − 25-s − 1.85·29-s + 0.718·31-s − 0.328·37-s − 1.56·41-s + 0.609·43-s − 1.75·47-s + 0.824·53-s + 0.260·59-s − 0.949·71-s + 0.702·73-s + 0.900·79-s + 1.53·83-s − 1.48·89-s + 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162288 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.687721737\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.687721737\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 2 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 14 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−13.31888622223207, −12.72989684090412, −12.29130090553936, −11.78704306001226, −11.52116748820753, −10.92556773387436, −10.33145564116688, −9.887314360740399, −9.522163950350100, −8.938351619972000, −8.456919482319139, −7.949305054298049, −7.496750533083307, −6.873010845414621, −6.350366448351614, −6.007562436965736, −5.335182958605284, −4.892180448787572, −3.982537121157146, −3.836038887708094, −3.275629773302998, −2.413568312062768, −1.751694465775790, −1.357913775499097, −0.3742548448023080,
0.3742548448023080, 1.357913775499097, 1.751694465775790, 2.413568312062768, 3.275629773302998, 3.836038887708094, 3.982537121157146, 4.892180448787572, 5.335182958605284, 6.007562436965736, 6.350366448351614, 6.873010845414621, 7.496750533083307, 7.949305054298049, 8.456919482319139, 8.938351619972000, 9.522163950350100, 9.887314360740399, 10.33145564116688, 10.92556773387436, 11.52116748820753, 11.78704306001226, 12.29130090553936, 12.72989684090412, 13.31888622223207
