L(s) = 1 | − 2-s + 4-s − 5-s − 7-s − 8-s + 10-s + 14-s + 16-s − 20-s + 23-s + 25-s − 28-s + 29-s − 32-s + 35-s + 40-s + 41-s + 2·43-s − 46-s + 47-s − 50-s + 56-s − 58-s − 61-s + 64-s − 67-s − 70-s + ⋯ |
L(s) = 1 | − 2-s + 4-s − 5-s − 7-s − 8-s + 10-s + 14-s + 16-s − 20-s + 23-s + 25-s − 28-s + 29-s − 32-s + 35-s + 40-s + 41-s + 2·43-s − 46-s + 47-s − 50-s + 56-s − 58-s − 61-s + 64-s − 67-s − 70-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5146376755\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5146376755\) |
\(L(1)\) |
|
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 | \( 1 \) |
| 5 | \( 1 + T \) |
good | 7 | \( 1 + T + T^{2} \) |
| 11 | \( ( 1 - T )( 1 + T ) \) |
| 13 | \( ( 1 - T )( 1 + T ) \) |
| 17 | \( ( 1 - T )( 1 + T ) \) |
| 19 | \( ( 1 - T )( 1 + T ) \) |
| 23 | \( 1 - T + T^{2} \) |
| 29 | \( 1 - T + T^{2} \) |
| 31 | \( ( 1 - T )( 1 + T ) \) |
| 37 | \( ( 1 - T )( 1 + T ) \) |
| 41 | \( 1 - T + T^{2} \) |
| 43 | \( ( 1 - T )^{2} \) |
| 47 | \( 1 - T + T^{2} \) |
| 53 | \( ( 1 - T )( 1 + T ) \) |
| 59 | \( ( 1 - T )( 1 + T ) \) |
| 61 | \( 1 + T + T^{2} \) |
| 67 | \( 1 + T + T^{2} \) |
| 71 | \( ( 1 - T )( 1 + T ) \) |
| 73 | \( ( 1 - T )( 1 + T ) \) |
| 79 | \( ( 1 - T )( 1 + T ) \) |
| 83 | \( 1 - T + T^{2} \) |
| 89 | \( 1 - T + T^{2} \) |
| 97 | \( ( 1 - T )( 1 + T ) \) |
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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
−9.328267571207584084177432898244, −8.971122138193663323119465315687, −7.976633669840458829407452138761, −7.33827256687520353584847305090, −6.63628129898705508292031270019, −5.80961598868882650276574263201, −4.46430431860143022442148930394, −3.36418788302137685670479392342, −2.62124025035193466925067073168, −0.849726787406769743318117101218,
0.849726787406769743318117101218, 2.62124025035193466925067073168, 3.36418788302137685670479392342, 4.46430431860143022442148930394, 5.80961598868882650276574263201, 6.63628129898705508292031270019, 7.33827256687520353584847305090, 7.976633669840458829407452138761, 8.971122138193663323119465315687, 9.328267571207584084177432898244