| L(s) = 1 | − 6·3-s + 6·5-s − 16·7-s + 27·9-s − 22·11-s − 22·13-s − 36·15-s − 22·17-s − 62·19-s + 96·21-s + 210·23-s − 86·25-s − 108·27-s + 214·29-s + 328·31-s + 132·33-s − 96·35-s + 200·37-s + 132·39-s + 114·41-s − 606·43-s + 162·45-s + 382·47-s + 54·49-s + 132·51-s + 2·53-s − 132·55-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.536·5-s − 0.863·7-s + 9-s − 0.603·11-s − 0.469·13-s − 0.619·15-s − 0.313·17-s − 0.748·19-s + 0.997·21-s + 1.90·23-s − 0.687·25-s − 0.769·27-s + 1.37·29-s + 1.90·31-s + 0.696·33-s − 0.463·35-s + 0.888·37-s + 0.541·39-s + 0.434·41-s − 2.14·43-s + 0.536·45-s + 1.18·47-s + 0.157·49-s + 0.362·51-s + 0.00518·53-s − 0.323·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4460544 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4460544 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 + p T )^{2} \) |
| 11 | $C_1$ | \( ( 1 + p T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 - 6 T + 122 T^{2} - 6 p^{3} T^{3} + p^{6} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 16 T + 202 T^{2} + 16 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 22 T + 4378 T^{2} + 22 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 22 T + 8714 T^{2} + 22 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 62 T + 13446 T^{2} + 62 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 210 T + 31934 T^{2} - 210 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 214 T + 53514 T^{2} - 214 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 328 T + 84286 T^{2} - 328 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 200 T + 110758 T^{2} - 200 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 114 T + 25874 T^{2} - 114 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 606 T + 211230 T^{2} + 606 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 382 T + 183710 T^{2} - 382 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 2 T + 182538 T^{2} - 2 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 1084 T + 703974 T^{2} + 1084 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 354 T + 70866 T^{2} - 354 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 456 T + 644742 T^{2} - 456 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 34 T + 616238 T^{2} - 34 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 112 T + 109870 T^{2} + 112 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 820 T + 1099378 T^{2} + 820 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 1012 T + 478422 T^{2} + 1012 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 1036 T + 1658534 T^{2} - 1036 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 1308 T + 2217990 T^{2} - 1308 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−8.579543922565863800575491334426, −8.170175762972655215167891453312, −7.70402469988888383477242548892, −7.25474025076188390527265916867, −6.82053636505028428230313031556, −6.51720387714690591690413716425, −6.18984533869279141511714632799, −5.95775631925830384795623087339, −5.27846093357250755201574045954, −5.02957630687806871130984165753, −4.47796898347901354970849647735, −4.42302759315703619154937400650, −3.51254979866984090404094790285, −3.07332731810258926105659995186, −2.51144709292696402862363764563, −2.26289862778333520701665699965, −1.13718427157199356669876605765, −1.10031837398796014371417266506, 0, 0,
1.10031837398796014371417266506, 1.13718427157199356669876605765, 2.26289862778333520701665699965, 2.51144709292696402862363764563, 3.07332731810258926105659995186, 3.51254979866984090404094790285, 4.42302759315703619154937400650, 4.47796898347901354970849647735, 5.02957630687806871130984165753, 5.27846093357250755201574045954, 5.95775631925830384795623087339, 6.18984533869279141511714632799, 6.51720387714690591690413716425, 6.82053636505028428230313031556, 7.25474025076188390527265916867, 7.70402469988888383477242548892, 8.170175762972655215167891453312, 8.579543922565863800575491334426
