L(s) = 1 | − 5-s + 2·7-s + 2·11-s + 2·13-s − 6·19-s + 6·23-s + 25-s − 6·29-s − 8·31-s − 2·35-s + 6·37-s + 6·41-s − 43-s − 2·47-s − 3·49-s − 14·53-s − 2·55-s + 6·59-s + 4·61-s − 2·65-s + 4·67-s + 8·71-s + 8·73-s + 4·77-s − 8·79-s − 8·83-s − 2·89-s + ⋯ |
L(s) = 1 | − 0.447·5-s + 0.755·7-s + 0.603·11-s + 0.554·13-s − 1.37·19-s + 1.25·23-s + 1/5·25-s − 1.11·29-s − 1.43·31-s − 0.338·35-s + 0.986·37-s + 0.937·41-s − 0.152·43-s − 0.291·47-s − 3/7·49-s − 1.92·53-s − 0.269·55-s + 0.781·59-s + 0.512·61-s − 0.248·65-s + 0.488·67-s + 0.949·71-s + 0.936·73-s + 0.455·77-s − 0.900·79-s − 0.878·83-s − 0.211·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 123840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 123840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 43 | \( 1 + T \) |
good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 + 14 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−13.85204576718527, −13.06587487509479, −12.79606156729657, −12.56780857379529, −11.54657418910036, −11.38559808670748, −10.95855260541389, −10.66932485294134, −9.798291296539552, −9.214894857795126, −9.004520076535118, −8.230602297096054, −8.017717906110110, −7.376286232126800, −6.780148033738200, −6.416331510844233, −5.681459663745799, −5.217376361225449, −4.544012966279755, −4.093544463582015, −3.612718589442103, −2.940826452170771, −2.119233135229629, −1.606636270324658, −0.9043380751523418, 0,
0.9043380751523418, 1.606636270324658, 2.119233135229629, 2.940826452170771, 3.612718589442103, 4.093544463582015, 4.544012966279755, 5.217376361225449, 5.681459663745799, 6.416331510844233, 6.780148033738200, 7.376286232126800, 8.017717906110110, 8.230602297096054, 9.004520076535118, 9.214894857795126, 9.798291296539552, 10.66932485294134, 10.95855260541389, 11.38559808670748, 11.54657418910036, 12.56780857379529, 12.79606156729657, 13.06587487509479, 13.85204576718527