L(s) = 1 | + 5-s − 4·7-s − 2·13-s + 5·17-s + 8·19-s − 5·23-s + 25-s − 6·29-s − 9·31-s − 4·35-s − 4·37-s − 6·41-s + 6·43-s + 13·47-s + 9·49-s − 9·53-s + 10·59-s + 11·61-s − 2·65-s + 12·67-s − 8·71-s − 4·73-s + 5·79-s − 8·83-s + 5·85-s + 8·89-s + 8·91-s + ⋯ |
L(s) = 1 | + 0.447·5-s − 1.51·7-s − 0.554·13-s + 1.21·17-s + 1.83·19-s − 1.04·23-s + 1/5·25-s − 1.11·29-s − 1.61·31-s − 0.676·35-s − 0.657·37-s − 0.937·41-s + 0.914·43-s + 1.89·47-s + 9/7·49-s − 1.23·53-s + 1.30·59-s + 1.40·61-s − 0.248·65-s + 1.46·67-s − 0.949·71-s − 0.468·73-s + 0.562·79-s − 0.878·83-s + 0.542·85-s + 0.847·89-s + 0.838·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21780 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21780 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 11 | \( 1 \) |
good | 7 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + 5 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 9 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 - 13 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 - 11 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 5 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 - 8 T + p T^{2} \) |
| 97 | \( 1 + p T^{2} \) |
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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
−15.95771094092056, −15.44829297258835, −14.51870476841951, −14.22797054606080, −13.71481685582306, −12.92329724666440, −12.74255842187274, −12.01917547105925, −11.63280536096701, −10.75670012631581, −9.986299048561770, −9.886576907648280, −9.295138499316067, −8.795847049479891, −7.712828104276213, −7.392745777811692, −6.825997789406254, −5.983204365382665, −5.541676250719310, −5.138171305726312, −3.753227570641452, −3.622045528813126, −2.790299405676194, −2.027154230084821, −1.007998711877648, 0,
1.007998711877648, 2.027154230084821, 2.790299405676194, 3.622045528813126, 3.753227570641452, 5.138171305726312, 5.541676250719310, 5.983204365382665, 6.825997789406254, 7.392745777811692, 7.712828104276213, 8.795847049479891, 9.295138499316067, 9.886576907648280, 9.986299048561770, 10.75670012631581, 11.63280536096701, 12.01917547105925, 12.74255842187274, 12.92329724666440, 13.71481685582306, 14.22797054606080, 14.51870476841951, 15.44829297258835, 15.95771094092056