| L(s) = 1 | − 2·25-s − 16·31-s + 16·49-s − 48·71-s + 16·79-s − 24·89-s + 20·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 52·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + ⋯ |
| L(s) = 1 | − 2/5·25-s − 2.87·31-s + 16/7·49-s − 5.69·71-s + 1.80·79-s − 2.54·89-s + 1.81·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 4·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s + 0.0663·227-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.543148577\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.543148577\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 5 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| good | 7 | $C_2^2$ | \( ( 1 - 8 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 17 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2}( 1 + 8 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 40 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 37 | $C_2^2$ | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 + 68 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 40 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 + 74 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 - 110 T^{2} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 + 116 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{4} \) |
| 73 | $C_2^2$ | \( ( 1 - 122 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 + 68 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
| 97 | $C_2^2$ | \( ( 1 - 170 T^{2} + p^{2} T^{4} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.87061788467267517973239168148, −6.78494370306904827676185864110, −6.41390258108764009558605608021, −5.81102722183507265951007400768, −5.79854871834206405341114892607, −5.76032087479538221669219931509, −5.75388813348236696055191699348, −5.42316349609168558919579582452, −4.94573414617589409908264763140, −4.82230991139684095967082334592, −4.46820265929862383138278898878, −4.44146982229831115454598954067, −3.99498284015343970927303072543, −3.84752166341094296791518520345, −3.75142524380012591260963863829, −3.22935701014259046564819050188, −3.08486870828165252512039467054, −2.89956417579053420112883044089, −2.50284544861769443092291496195, −2.17894637240117982510833337269, −1.90100774613103420509365755861, −1.50822105895180931130633762586, −1.45264234883724348238817836755, −0.72388541388547532430577311565, −0.28780089838918168730043399606,
0.28780089838918168730043399606, 0.72388541388547532430577311565, 1.45264234883724348238817836755, 1.50822105895180931130633762586, 1.90100774613103420509365755861, 2.17894637240117982510833337269, 2.50284544861769443092291496195, 2.89956417579053420112883044089, 3.08486870828165252512039467054, 3.22935701014259046564819050188, 3.75142524380012591260963863829, 3.84752166341094296791518520345, 3.99498284015343970927303072543, 4.44146982229831115454598954067, 4.46820265929862383138278898878, 4.82230991139684095967082334592, 4.94573414617589409908264763140, 5.42316349609168558919579582452, 5.75388813348236696055191699348, 5.76032087479538221669219931509, 5.79854871834206405341114892607, 5.81102722183507265951007400768, 6.41390258108764009558605608021, 6.78494370306904827676185864110, 6.87061788467267517973239168148
