L(s) = 1 | + 5-s − 2·7-s + 11-s + 5·13-s + 3·17-s + 6·19-s − 5·23-s + 25-s − 7·31-s − 2·35-s + 8·37-s − 5·41-s − 43-s + 8·47-s − 3·49-s − 9·53-s + 55-s − 4·59-s − 8·61-s + 5·65-s − 11·67-s + 14·71-s − 4·73-s − 2·77-s + 16·79-s + 83-s + 3·85-s + ⋯ |
L(s) = 1 | + 0.447·5-s − 0.755·7-s + 0.301·11-s + 1.38·13-s + 0.727·17-s + 1.37·19-s − 1.04·23-s + 1/5·25-s − 1.25·31-s − 0.338·35-s + 1.31·37-s − 0.780·41-s − 0.152·43-s + 1.16·47-s − 3/7·49-s − 1.23·53-s + 0.134·55-s − 0.520·59-s − 1.02·61-s + 0.620·65-s − 1.34·67-s + 1.66·71-s − 0.468·73-s − 0.227·77-s + 1.80·79-s + 0.109·83-s + 0.325·85-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 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 5 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 11 T + p T^{2} \) |
| 71 | \( 1 - 14 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 - T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 17 T + p T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.75748536985703, −13.42746774983646, −12.81266888627148, −12.33008861057828, −11.95522396884399, −11.29695524322706, −10.87981414957065, −10.39900300946777, −9.698926276732647, −9.470805311752725, −9.096390676983168, −8.312307864872886, −7.904505431860805, −7.359197106637136, −6.736499360907265, −6.152674668553591, −5.875086307538565, −5.384089171217194, −4.638422994968420, −3.936347227165679, −3.382112007705180, −3.121869138674570, −2.207330124974103, −1.468478748680458, −1.004233886268762, 0,
1.004233886268762, 1.468478748680458, 2.207330124974103, 3.121869138674570, 3.382112007705180, 3.936347227165679, 4.638422994968420, 5.384089171217194, 5.875086307538565, 6.152674668553591, 6.736499360907265, 7.359197106637136, 7.904505431860805, 8.312307864872886, 9.096390676983168, 9.470805311752725, 9.698926276732647, 10.39900300946777, 10.87981414957065, 11.29695524322706, 11.95522396884399, 12.33008861057828, 12.81266888627148, 13.42746774983646, 13.75748536985703