| L(s) = 1 | + 3-s − 3·5-s + 3·7-s − 2·9-s − 3·15-s − 3·17-s + 6·19-s + 3·21-s − 6·23-s + 4·25-s − 5·27-s − 9·35-s − 3·37-s − 43-s + 6·45-s + 3·47-s + 2·49-s − 3·51-s − 6·53-s + 6·57-s − 6·59-s − 8·61-s − 6·63-s + 12·67-s − 6·69-s − 15·71-s − 6·73-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.34·5-s + 1.13·7-s − 2/3·9-s − 0.774·15-s − 0.727·17-s + 1.37·19-s + 0.654·21-s − 1.25·23-s + 4/5·25-s − 0.962·27-s − 1.52·35-s − 0.493·37-s − 0.152·43-s + 0.894·45-s + 0.437·47-s + 2/7·49-s − 0.420·51-s − 0.824·53-s + 0.794·57-s − 0.781·59-s − 1.02·61-s − 0.755·63-s + 1.46·67-s − 0.722·69-s − 1.78·71-s − 0.702·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2704 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2704 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 + 3 T + p T^{2} \) |
| 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 15 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
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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
−8.400975960193558050461238177888, −7.71760859062717889479366288712, −7.39367722152941399853201814879, −6.13608294207511231988617194090, −5.17971243854197951089544579950, −4.38415404539054737496191815769, −3.63219307944314459870242767288, −2.76087545550431720024259429471, −1.58794048017917856778262314054, 0,
1.58794048017917856778262314054, 2.76087545550431720024259429471, 3.63219307944314459870242767288, 4.38415404539054737496191815769, 5.17971243854197951089544579950, 6.13608294207511231988617194090, 7.39367722152941399853201814879, 7.71760859062717889479366288712, 8.400975960193558050461238177888
