L(s) = 1 | − 3-s − 2·7-s + 9-s + 11-s − 17-s + 8·19-s + 2·21-s − 4·23-s − 27-s − 6·29-s + 4·31-s − 33-s + 2·37-s + 10·43-s + 8·47-s − 3·49-s + 51-s + 6·53-s − 8·57-s − 6·59-s − 2·61-s − 2·63-s − 10·67-s + 4·69-s − 6·71-s + 8·73-s − 2·77-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.755·7-s + 1/3·9-s + 0.301·11-s − 0.242·17-s + 1.83·19-s + 0.436·21-s − 0.834·23-s − 0.192·27-s − 1.11·29-s + 0.718·31-s − 0.174·33-s + 0.328·37-s + 1.52·43-s + 1.16·47-s − 3/7·49-s + 0.140·51-s + 0.824·53-s − 1.05·57-s − 0.781·59-s − 0.256·61-s − 0.251·63-s − 1.22·67-s + 0.481·69-s − 0.712·71-s + 0.936·73-s − 0.227·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 224400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 \) |
| 11 | \( 1 - T \) |
| 17 | \( 1 + T \) |
good | 7 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−13.27324772553929, −12.52629651982297, −12.24058120539826, −11.89516570208372, −11.32247714821198, −10.92425030254711, −10.40296625078716, −9.861729371538894, −9.429299647617498, −9.232910189453189, −8.503756911414127, −7.849166898812487, −7.365315199819952, −7.112451434747806, −6.378699608918448, −5.855575035865978, −5.735898164141650, −4.947725893824891, −4.442811570148703, −3.827052999191126, −3.398180628281030, −2.741849282848437, −2.144207267381293, −1.311549343541792, −0.7783649645307803, 0,
0.7783649645307803, 1.311549343541792, 2.144207267381293, 2.741849282848437, 3.398180628281030, 3.827052999191126, 4.442811570148703, 4.947725893824891, 5.735898164141650, 5.855575035865978, 6.378699608918448, 7.112451434747806, 7.365315199819952, 7.849166898812487, 8.503756911414127, 9.232910189453189, 9.429299647617498, 9.861729371538894, 10.40296625078716, 10.92425030254711, 11.32247714821198, 11.89516570208372, 12.24058120539826, 12.52629651982297, 13.27324772553929