| L(s) = 1 | − 5-s + 7-s − 4·11-s + 2·13-s + 8·17-s − 6·19-s − 4·23-s + 25-s − 2·29-s + 4·31-s − 35-s − 8·37-s + 10·41-s − 10·43-s + 10·47-s + 49-s + 2·53-s + 4·55-s + 4·59-s + 4·61-s − 2·65-s − 10·67-s − 6·71-s + 6·73-s − 4·77-s − 8·79-s + 8·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.377·7-s − 1.20·11-s + 0.554·13-s + 1.94·17-s − 1.37·19-s − 0.834·23-s + 1/5·25-s − 0.371·29-s + 0.718·31-s − 0.169·35-s − 1.31·37-s + 1.56·41-s − 1.52·43-s + 1.45·47-s + 1/7·49-s + 0.274·53-s + 0.539·55-s + 0.520·59-s + 0.512·61-s − 0.248·65-s − 1.22·67-s − 0.712·71-s + 0.702·73-s − 0.455·77-s − 0.900·79-s + 0.878·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| good | 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 8 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−16.88033337772730, −16.25221402298082, −15.84154183878662, −15.16727541425512, −14.71185900839610, −14.06872344359688, −13.47613996718940, −12.83739266358698, −12.21563046517785, −11.84148168506735, −10.97271168252681, −10.43751759457465, −10.11809270192935, −9.174898964645487, −8.330999685457580, −8.078660191615066, −7.462371450304908, −6.689705751968553, −5.734527682772159, −5.435598486557134, −4.464931999587222, −3.836456150662290, −3.032161677770512, −2.196741765561287, −1.179867541116159, 0,
1.179867541116159, 2.196741765561287, 3.032161677770512, 3.836456150662290, 4.464931999587222, 5.435598486557134, 5.734527682772159, 6.689705751968553, 7.462371450304908, 8.078660191615066, 8.330999685457580, 9.174898964645487, 10.11809270192935, 10.43751759457465, 10.97271168252681, 11.84148168506735, 12.21563046517785, 12.83739266358698, 13.47613996718940, 14.06872344359688, 14.71185900839610, 15.16727541425512, 15.84154183878662, 16.25221402298082, 16.88033337772730
