| L(s) = 1 | − 2·5-s − 7-s − 3·9-s + 4·11-s + 2·17-s − 4·19-s + 4·23-s − 25-s + 2·29-s + 2·35-s + 2·37-s + 2·41-s + 4·43-s + 6·45-s − 8·47-s + 49-s + 2·53-s − 8·55-s + 4·59-s − 14·61-s + 3·63-s − 12·67-s + 8·71-s − 14·73-s − 4·77-s + 12·79-s + 9·81-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 0.377·7-s − 9-s + 1.20·11-s + 0.485·17-s − 0.917·19-s + 0.834·23-s − 1/5·25-s + 0.371·29-s + 0.338·35-s + 0.328·37-s + 0.312·41-s + 0.609·43-s + 0.894·45-s − 1.16·47-s + 1/7·49-s + 0.274·53-s − 1.07·55-s + 0.520·59-s − 1.79·61-s + 0.377·63-s − 1.46·67-s + 0.949·71-s − 1.63·73-s − 0.455·77-s + 1.35·79-s + 81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75712 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 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 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 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
−14.37796198882090, −13.89777676749566, −13.31581218593354, −12.67973182580703, −12.25304176453250, −11.80884271234814, −11.31771593813609, −11.01220159573242, −10.31775550237983, −9.734054823961064, −9.094566451636711, −8.767282688248230, −8.269640955601634, −7.632943632280902, −7.193438672296683, −6.452264372390881, −6.127240380940390, −5.545514675445481, −4.653581511956927, −4.314299689154804, −3.543981909273187, −3.202720660430773, −2.495902709976859, −1.607692229794035, −0.7786653953247407, 0,
0.7786653953247407, 1.607692229794035, 2.495902709976859, 3.202720660430773, 3.543981909273187, 4.314299689154804, 4.653581511956927, 5.545514675445481, 6.127240380940390, 6.452264372390881, 7.193438672296683, 7.632943632280902, 8.269640955601634, 8.767282688248230, 9.094566451636711, 9.734054823961064, 10.31775550237983, 11.01220159573242, 11.31771593813609, 11.80884271234814, 12.25304176453250, 12.67973182580703, 13.31581218593354, 13.89777676749566, 14.37796198882090
