L(s) = 1 | + 3-s + 9-s − 2·19-s + 25-s + 27-s − 2·43-s − 49-s − 2·57-s + 2·67-s − 2·73-s + 75-s + 81-s − 2·97-s + ⋯ |
L(s) = 1 | + 3-s + 9-s − 2·19-s + 25-s + 27-s − 2·43-s − 49-s − 2·57-s + 2·67-s − 2·73-s + 75-s + 81-s − 2·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.308874648\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.308874648\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
good | 5 | \( ( 1 - T )( 1 + T ) \) |
| 7 | \( 1 + T^{2} \) |
| 11 | \( ( 1 - T )( 1 + T ) \) |
| 13 | \( 1 + T^{2} \) |
| 17 | \( ( 1 - T )( 1 + T ) \) |
| 19 | \( ( 1 + T )^{2} \) |
| 23 | \( ( 1 - T )( 1 + T ) \) |
| 29 | \( ( 1 - T )( 1 + T ) \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( 1 + T^{2} \) |
| 41 | \( ( 1 - T )( 1 + T ) \) |
| 43 | \( ( 1 + T )^{2} \) |
| 47 | \( ( 1 - T )( 1 + T ) \) |
| 53 | \( ( 1 - T )( 1 + T ) \) |
| 59 | \( ( 1 - T )( 1 + T ) \) |
| 61 | \( 1 + T^{2} \) |
| 67 | \( ( 1 - T )^{2} \) |
| 71 | \( ( 1 - T )( 1 + T ) \) |
| 73 | \( ( 1 + T )^{2} \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( ( 1 - T )( 1 + T ) \) |
| 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
−10.39102662824977119738323901130, −9.642217198254878916592501148423, −8.610740553163108063340259236730, −8.269999815519138953654602968242, −7.07481545445483216171663500370, −6.38352310888271867017871585322, −4.93102551732940022120334392274, −4.02671025344536349797829606424, −2.93433446474655514678138565351, −1.80861110516981843613056691839,
1.80861110516981843613056691839, 2.93433446474655514678138565351, 4.02671025344536349797829606424, 4.93102551732940022120334392274, 6.38352310888271867017871585322, 7.07481545445483216171663500370, 8.269999815519138953654602968242, 8.610740553163108063340259236730, 9.642217198254878916592501148423, 10.39102662824977119738323901130