| L(s) = 1 | + 3-s + 3·5-s + 5·7-s − 2·11-s + 7·13-s + 3·15-s − 6·17-s − 2·19-s + 5·21-s − 6·23-s + 2·25-s + 2·27-s + 9·29-s + 17·31-s − 2·33-s + 15·35-s + 16·37-s + 7·39-s + 3·41-s − 43-s + 10·49-s − 6·51-s + 6·53-s − 6·55-s − 2·57-s + 4·61-s + 21·65-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.34·5-s + 1.88·7-s − 0.603·11-s + 1.94·13-s + 0.774·15-s − 1.45·17-s − 0.458·19-s + 1.09·21-s − 1.25·23-s + 2/5·25-s + 0.384·27-s + 1.67·29-s + 3.05·31-s − 0.348·33-s + 2.53·35-s + 2.63·37-s + 1.12·39-s + 0.468·41-s − 0.152·43-s + 10/7·49-s − 0.840·51-s + 0.824·53-s − 0.809·55-s − 0.264·57-s + 0.512·61-s + 2.60·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11182336 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11182336 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.668808474\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.668808474\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 + T )^{2} \) |
| 19 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 - T + T^{2} - p T^{3} + p^{2} T^{4} \) |
| 5 | $D_{4}$ | \( 1 - 3 T + 7 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 5 T + 15 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 7 T + 33 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 6 T + 22 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 6 T + 34 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 9 T + 73 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 17 T + 129 T^{2} - 17 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 41 | $D_{4}$ | \( 1 - 3 T + 79 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + T + 39 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 6 T + 94 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 67 | $D_{4}$ | \( 1 - 5 T + 135 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 9 T + 115 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 10 T + 150 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 2 T + 138 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 3 T + 163 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 18 T + 238 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.486807185866756808031225617644, −8.400399505010108897872101436719, −8.142490351811044314131412856339, −8.097714889068247609037348950572, −7.51981563924045934483629141209, −6.61833617184988910411403824790, −6.46654781869563031625894102157, −6.40355553839369254166287429616, −5.74906222594953493391441377177, −5.49619087486239944381125007344, −4.92906925361413416353411607223, −4.48925273948586296338615918432, −4.23917730801176553698487534301, −3.99105368593266749617840023358, −2.99004905331838308003568619485, −2.63289081188703848382747190153, −2.26052903083815587738649227849, −1.94377543123514766152516822837, −1.10605713476119079480259838507, −0.983137180192912163317366735551,
0.983137180192912163317366735551, 1.10605713476119079480259838507, 1.94377543123514766152516822837, 2.26052903083815587738649227849, 2.63289081188703848382747190153, 2.99004905331838308003568619485, 3.99105368593266749617840023358, 4.23917730801176553698487534301, 4.48925273948586296338615918432, 4.92906925361413416353411607223, 5.49619087486239944381125007344, 5.74906222594953493391441377177, 6.40355553839369254166287429616, 6.46654781869563031625894102157, 6.61833617184988910411403824790, 7.51981563924045934483629141209, 8.097714889068247609037348950572, 8.142490351811044314131412856339, 8.400399505010108897872101436719, 8.486807185866756808031225617644
