L(s) = 1 | − 11·7-s + 23·13-s + 37·19-s + 25·25-s + 46·31-s − 73·37-s + 22·43-s + 72·49-s + 47·61-s + 13·67-s + 143·73-s − 11·79-s − 253·91-s − 169·97-s + 157·103-s − 214·109-s + ⋯ |
L(s) = 1 | − 1.57·7-s + 1.76·13-s + 1.94·19-s + 25-s + 1.48·31-s − 1.97·37-s + 0.511·43-s + 1.46·49-s + 0.770·61-s + 0.194·67-s + 1.95·73-s − 0.139·79-s − 2.78·91-s − 1.74·97-s + 1.52·103-s − 1.96·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.565956920\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.565956920\) |
\(L(2)\) |
|
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 | \( ( 1 - p T )( 1 + p T ) \) |
| 7 | \( 1 + 11 T + p^{2} T^{2} \) |
| 11 | \( ( 1 - p T )( 1 + p T ) \) |
| 13 | \( 1 - 23 T + p^{2} T^{2} \) |
| 17 | \( ( 1 - p T )( 1 + p T ) \) |
| 19 | \( 1 - 37 T + p^{2} T^{2} \) |
| 23 | \( ( 1 - p T )( 1 + p T ) \) |
| 29 | \( ( 1 - p T )( 1 + p T ) \) |
| 31 | \( 1 - 46 T + p^{2} T^{2} \) |
| 37 | \( 1 + 73 T + p^{2} T^{2} \) |
| 41 | \( ( 1 - p T )( 1 + p T ) \) |
| 43 | \( 1 - 22 T + p^{2} T^{2} \) |
| 47 | \( ( 1 - p T )( 1 + p T ) \) |
| 53 | \( ( 1 - p T )( 1 + p T ) \) |
| 59 | \( ( 1 - p T )( 1 + p T ) \) |
| 61 | \( 1 - 47 T + p^{2} T^{2} \) |
| 67 | \( 1 - 13 T + p^{2} T^{2} \) |
| 71 | \( ( 1 - p T )( 1 + p T ) \) |
| 73 | \( 1 - 143 T + p^{2} T^{2} \) |
| 79 | \( 1 + 11 T + p^{2} T^{2} \) |
| 83 | \( ( 1 - p T )( 1 + p T ) \) |
| 89 | \( ( 1 - p T )( 1 + p T ) \) |
| 97 | \( 1 + 169 T + p^{2} T^{2} \) |
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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
−10.86076580905130567718551924642, −9.989623032380754154401643151522, −9.203468854018199565723952911291, −8.331287839344309215851422522644, −7.01891639391506344634181018241, −6.32241251273778247419460637694, −5.32303470874439342476960852367, −3.71899369818720426939136696502, −3.01597961007327540116277505066, −0.978565515671405865293034079298,
0.978565515671405865293034079298, 3.01597961007327540116277505066, 3.71899369818720426939136696502, 5.32303470874439342476960852367, 6.32241251273778247419460637694, 7.01891639391506344634181018241, 8.331287839344309215851422522644, 9.203468854018199565723952911291, 9.989623032380754154401643151522, 10.86076580905130567718551924642