L(s) = 1 | + 2·5-s + 3·25-s − 2·29-s + 49-s − 2·53-s − 2·73-s + 2·97-s − 2·101-s + ⋯ |
L(s) = 1 | + 2·5-s + 3·25-s − 2·29-s + 49-s − 2·53-s − 2·73-s + 2·97-s − 2·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.677847420\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.677847420\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( ( 1 - T )^{2} \) |
| 7 | \( ( 1 - T )( 1 + T ) \) |
| 11 | \( ( 1 - T )( 1 + T ) \) |
| 13 | \( 1 + T^{2} \) |
| 17 | \( 1 + T^{2} \) |
| 19 | \( ( 1 - T )( 1 + T ) \) |
| 23 | \( ( 1 - T )( 1 + T ) \) |
| 29 | \( ( 1 + T )^{2} \) |
| 31 | \( ( 1 - T )( 1 + T ) \) |
| 37 | \( 1 + T^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( ( 1 - T )( 1 + T ) \) |
| 47 | \( ( 1 - T )( 1 + T ) \) |
| 53 | \( ( 1 + T )^{2} \) |
| 59 | \( ( 1 - T )( 1 + T ) \) |
| 61 | \( 1 + T^{2} \) |
| 67 | \( ( 1 - T )( 1 + T ) \) |
| 71 | \( ( 1 - T )( 1 + T ) \) |
| 73 | \( ( 1 + T )^{2} \) |
| 79 | \( ( 1 - T )( 1 + T ) \) |
| 83 | \( ( 1 - T )( 1 + T ) \) |
| 89 | \( 1 + T^{2} \) |
| 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
−9.322003184113943316768415254980, −8.686248992562608132329204370109, −7.55841498540233778562255189232, −6.73344883833152527271659122388, −5.92670147167541228297441004653, −5.48771189097825729323690058692, −4.55601665929781424215110545537, −3.25373691694389316726401599253, −2.25339764419376934619980919755, −1.48378970333674093865521528458,
1.48378970333674093865521528458, 2.25339764419376934619980919755, 3.25373691694389316726401599253, 4.55601665929781424215110545537, 5.48771189097825729323690058692, 5.92670147167541228297441004653, 6.73344883833152527271659122388, 7.55841498540233778562255189232, 8.686248992562608132329204370109, 9.322003184113943316768415254980