| L(s) = 1 | − 5-s − 7-s − 4·11-s − 6·13-s − 4·17-s + 2·19-s + 8·23-s + 25-s + 6·29-s + 4·31-s + 35-s + 8·37-s + 2·41-s − 6·43-s − 6·47-s + 49-s + 2·53-s + 4·55-s − 4·59-s + 6·65-s − 14·67-s − 2·71-s + 6·73-s + 4·77-s + 8·79-s + 8·83-s + 4·85-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.377·7-s − 1.20·11-s − 1.66·13-s − 0.970·17-s + 0.458·19-s + 1.66·23-s + 1/5·25-s + 1.11·29-s + 0.718·31-s + 0.169·35-s + 1.31·37-s + 0.312·41-s − 0.914·43-s − 0.875·47-s + 1/7·49-s + 0.274·53-s + 0.539·55-s − 0.520·59-s + 0.744·65-s − 1.71·67-s − 0.237·71-s + 0.702·73-s + 0.455·77-s + 0.900·79-s + 0.878·83-s + 0.433·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 20160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + 14 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 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 | \( 1 - 18 T + p T^{2} \) |
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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
−15.82878507540639, −15.34414683698009, −14.97567057225168, −14.41840068551361, −13.59104983189204, −13.14187311158438, −12.75315660969798, −12.02282399937808, −11.65822003601967, −10.85295982834633, −10.45922965144085, −9.776044041381364, −9.317528214722778, −8.603411407563728, −7.922633074690133, −7.468156298903814, −6.867370272542470, −6.312363310060983, −5.354484684189507, −4.731482582061007, −4.537635922533045, −3.239065212828932, −2.818386015238048, −2.207964178553600, −0.8716262673883203, 0,
0.8716262673883203, 2.207964178553600, 2.818386015238048, 3.239065212828932, 4.537635922533045, 4.731482582061007, 5.354484684189507, 6.312363310060983, 6.867370272542470, 7.468156298903814, 7.922633074690133, 8.603411407563728, 9.317528214722778, 9.776044041381364, 10.45922965144085, 10.85295982834633, 11.65822003601967, 12.02282399937808, 12.75315660969798, 13.14187311158438, 13.59104983189204, 14.41840068551361, 14.97567057225168, 15.34414683698009, 15.82878507540639
