L(s) = 1 | − 3-s − 5-s − 2·9-s + 5·11-s + 15-s + 3·17-s − 2·19-s + 8·23-s + 25-s + 5·27-s − 5·29-s − 10·31-s − 5·33-s − 4·37-s − 6·41-s + 2·43-s + 2·45-s − 7·47-s − 3·51-s − 10·53-s − 5·55-s + 2·57-s − 10·59-s + 12·61-s + 2·67-s − 8·69-s − 2·73-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.447·5-s − 2/3·9-s + 1.50·11-s + 0.258·15-s + 0.727·17-s − 0.458·19-s + 1.66·23-s + 1/5·25-s + 0.962·27-s − 0.928·29-s − 1.79·31-s − 0.870·33-s − 0.657·37-s − 0.937·41-s + 0.304·43-s + 0.298·45-s − 1.02·47-s − 0.420·51-s − 1.37·53-s − 0.674·55-s + 0.264·57-s − 1.30·59-s + 1.53·61-s + 0.244·67-s − 0.963·69-s − 0.234·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 331240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 331240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6030953583\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6030953583\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| 13 | \( 1 \) |
good | 3 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + 7 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 - 12 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 7 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 8 T + p T^{2} \) |
| 97 | \( 1 - 17 T + p T^{2} \) |
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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
−12.44907236283765, −12.11454874805696, −11.65360032642564, −11.14201966122370, −11.04888165159926, −10.51434203280758, −9.784616679716853, −9.319328927372242, −8.982809468157372, −8.551351243885464, −8.014365031104680, −7.422193826070984, −6.871040797477437, −6.683746630202171, −6.001588789743688, −5.573900657883822, −5.050447680142252, −4.661460575826172, −3.883490401575430, −3.472077210872762, −3.155395999895012, −2.301942662841838, −1.497993320912746, −1.187272319937975, −0.2237214738826159,
0.2237214738826159, 1.187272319937975, 1.497993320912746, 2.301942662841838, 3.155395999895012, 3.472077210872762, 3.883490401575430, 4.661460575826172, 5.050447680142252, 5.573900657883822, 6.001588789743688, 6.683746630202171, 6.871040797477437, 7.422193826070984, 8.014365031104680, 8.551351243885464, 8.982809468157372, 9.319328927372242, 9.784616679716853, 10.51434203280758, 11.04888165159926, 11.14201966122370, 11.65360032642564, 12.11454874805696, 12.44907236283765