| L(s) = 1 | − 2-s − 3-s + 4-s + 6-s − 8-s + 9-s + 11-s − 12-s + 16-s + 2·17-s − 18-s − 22-s + 24-s − 25-s − 27-s − 32-s − 33-s − 2·34-s + 36-s − 2·41-s + 44-s − 48-s − 49-s + 50-s − 2·51-s + 54-s + 64-s + ⋯ |
| L(s) = 1 | − 2-s − 3-s + 4-s + 6-s − 8-s + 9-s + 11-s − 12-s + 16-s + 2·17-s − 18-s − 22-s + 24-s − 25-s − 27-s − 32-s − 33-s − 2·34-s + 36-s − 2·41-s + 44-s − 48-s − 49-s + 50-s − 2·51-s + 54-s + 64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 264 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 264 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4105270841\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4105270841\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| good | 5 | \( 1 + T^{2} \) |
| 7 | \( 1 + T^{2} \) |
| 13 | \( 1 + T^{2} \) |
| 17 | \( ( 1 - T )^{2} \) |
| 19 | \( ( 1 - T )( 1 + T ) \) |
| 23 | \( 1 + T^{2} \) |
| 29 | \( ( 1 - T )( 1 + T ) \) |
| 31 | \( ( 1 - T )( 1 + T ) \) |
| 37 | \( ( 1 - T )( 1 + T ) \) |
| 41 | \( ( 1 + T )^{2} \) |
| 43 | \( ( 1 - T )( 1 + T ) \) |
| 47 | \( 1 + T^{2} \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( ( 1 - T )( 1 + T ) \) |
| 61 | \( 1 + T^{2} \) |
| 67 | \( ( 1 - T )^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( ( 1 - T )( 1 + T ) \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( ( 1 + T )^{2} \) |
| 89 | \( ( 1 - T )( 1 + T ) \) |
| 97 | \( ( 1 + T )^{2} \) |
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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
−11.88451989650797107904539781928, −11.30238767356822738718639092233, −10.09069153626929297335704447362, −9.675979487524854751586233347964, −8.294767804190940999998727958881, −7.27754440837644720701665491952, −6.33764073208760212040798631320, −5.38007425380953534303618242882, −3.60975030390953103469712926046, −1.44474556202835591116477007583,
1.44474556202835591116477007583, 3.60975030390953103469712926046, 5.38007425380953534303618242882, 6.33764073208760212040798631320, 7.27754440837644720701665491952, 8.294767804190940999998727958881, 9.675979487524854751586233347964, 10.09069153626929297335704447362, 11.30238767356822738718639092233, 11.88451989650797107904539781928
