L(s) = 1 | + 2-s + 4-s + 7-s + 8-s − 11-s − 3·13-s + 14-s + 16-s − 4·17-s − 6·19-s − 22-s + 3·23-s − 3·26-s + 28-s + 6·29-s + 2·31-s + 32-s − 4·34-s + 7·37-s − 6·38-s − 2·41-s + 2·43-s − 44-s + 3·46-s − 7·47-s + 49-s − 3·52-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.377·7-s + 0.353·8-s − 0.301·11-s − 0.832·13-s + 0.267·14-s + 1/4·16-s − 0.970·17-s − 1.37·19-s − 0.213·22-s + 0.625·23-s − 0.588·26-s + 0.188·28-s + 1.11·29-s + 0.359·31-s + 0.176·32-s − 0.685·34-s + 1.15·37-s − 0.973·38-s − 0.312·41-s + 0.304·43-s − 0.150·44-s + 0.442·46-s − 1.02·47-s + 1/7·49-s − 0.416·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
good | 11 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + 7 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 13 T + p T^{2} \) |
| 61 | \( 1 + 11 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 - 13 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 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
−7.25491606316214754448486634080, −6.51850868686966780938136343599, −6.10533672959908880439611517492, −5.01724598541949405433937387484, −4.66855657350835386955561927073, −4.04729198277093112017901476816, −2.88749122822843856755394828281, −2.42773364032611524346206705427, −1.44222924188293535759327590219, 0,
1.44222924188293535759327590219, 2.42773364032611524346206705427, 2.88749122822843856755394828281, 4.04729198277093112017901476816, 4.66855657350835386955561927073, 5.01724598541949405433937387484, 6.10533672959908880439611517492, 6.51850868686966780938136343599, 7.25491606316214754448486634080