L(s) = 1 | − 3·5-s + 2·7-s + 6·11-s − 5·13-s − 3·17-s − 2·19-s + 6·23-s + 4·25-s − 3·29-s − 4·31-s − 6·35-s − 5·37-s − 6·41-s + 10·43-s − 3·49-s + 6·53-s − 18·55-s + 12·59-s − 5·61-s + 15·65-s − 2·67-s + 6·71-s − 73-s + 12·77-s − 10·79-s + 9·85-s − 3·89-s + ⋯ |
L(s) = 1 | − 1.34·5-s + 0.755·7-s + 1.80·11-s − 1.38·13-s − 0.727·17-s − 0.458·19-s + 1.25·23-s + 4/5·25-s − 0.557·29-s − 0.718·31-s − 1.01·35-s − 0.821·37-s − 0.937·41-s + 1.52·43-s − 3/7·49-s + 0.824·53-s − 2.42·55-s + 1.56·59-s − 0.640·61-s + 1.86·65-s − 0.244·67-s + 0.712·71-s − 0.117·73-s + 1.36·77-s − 1.12·79-s + 0.976·85-s − 0.317·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
good | 5 | \( 1 + 3 T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 5 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 5 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 3 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−7.79206679054421899349333907030, −7.03005553588351480002493974558, −6.84069557065893494797443679017, −5.54684314295137587398248812000, −4.70182016779532035923324176221, −4.16351938194235425334694993448, −3.51887106962682004302941081992, −2.35144079968225251690349015067, −1.29287038689760625081257235289, 0,
1.29287038689760625081257235289, 2.35144079968225251690349015067, 3.51887106962682004302941081992, 4.16351938194235425334694993448, 4.70182016779532035923324176221, 5.54684314295137587398248812000, 6.84069557065893494797443679017, 7.03005553588351480002493974558, 7.79206679054421899349333907030