| L(s) = 1 | + 4·3-s + 8·9-s + 4·11-s + 12·19-s − 10·25-s + 12·27-s + 16·33-s − 12·41-s + 12·43-s − 14·49-s + 48·57-s − 20·59-s + 12·67-s − 40·75-s + 23·81-s + 4·83-s + 32·99-s − 28·107-s − 36·113-s + 8·121-s − 48·123-s + 127-s + 48·129-s + 131-s + 137-s + 139-s − 56·147-s + ⋯ |
| L(s) = 1 | + 2.30·3-s + 8/3·9-s + 1.20·11-s + 2.75·19-s − 2·25-s + 2.30·27-s + 2.78·33-s − 1.87·41-s + 1.82·43-s − 2·49-s + 6.35·57-s − 2.60·59-s + 1.46·67-s − 4.61·75-s + 23/9·81-s + 0.439·83-s + 3.21·99-s − 2.70·107-s − 3.38·113-s + 8/11·121-s − 4.32·123-s + 0.0887·127-s + 4.22·129-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 4.61·147-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 262144 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 262144 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.486587907\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.486587907\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 12 T + 72 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 12 T + 72 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 20 T + 200 T^{2} + 20 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 12 T + 72 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 142 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 146 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 94 T^{2} + p^{2} T^{4} \) |
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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
−11.05896459907890179335266657182, −10.60417879617760738963477666130, −9.735398418577798032356123542837, −9.638579861377457242778016799469, −9.341735885037733232573126549883, −9.090498711873451502138258335141, −8.210915151021506824382224722761, −8.205758917225004819031133265734, −7.59261692385563935422250155263, −7.38422127119522909093918759373, −6.64112494412112710620413980932, −6.21675213726809133712376818377, −5.39130009475052051985370500645, −4.94269025013754950277331619420, −3.91165354404743641805998418181, −3.85552847733976615835611606496, −3.07421910463326912458503000960, −2.87309189232451863575736962880, −1.86215702794977185136290616837, −1.34921872461900061662499976057,
1.34921872461900061662499976057, 1.86215702794977185136290616837, 2.87309189232451863575736962880, 3.07421910463326912458503000960, 3.85552847733976615835611606496, 3.91165354404743641805998418181, 4.94269025013754950277331619420, 5.39130009475052051985370500645, 6.21675213726809133712376818377, 6.64112494412112710620413980932, 7.38422127119522909093918759373, 7.59261692385563935422250155263, 8.205758917225004819031133265734, 8.210915151021506824382224722761, 9.090498711873451502138258335141, 9.341735885037733232573126549883, 9.638579861377457242778016799469, 9.735398418577798032356123542837, 10.60417879617760738963477666130, 11.05896459907890179335266657182
