| L(s) = 1 | − 2·3-s − 2·7-s + 2·9-s + 2·13-s − 2·17-s − 8·19-s + 4·21-s + 10·23-s − 6·27-s + 10·37-s − 4·39-s + 12·41-s + 6·43-s + 14·47-s + 2·49-s + 4·51-s + 2·53-s + 16·57-s − 8·59-s + 4·61-s − 4·63-s − 14·67-s − 20·69-s − 18·73-s + 16·79-s + 11·81-s − 10·83-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 0.755·7-s + 2/3·9-s + 0.554·13-s − 0.485·17-s − 1.83·19-s + 0.872·21-s + 2.08·23-s − 1.15·27-s + 1.64·37-s − 0.640·39-s + 1.87·41-s + 0.914·43-s + 2.04·47-s + 2/7·49-s + 0.560·51-s + 0.274·53-s + 2.11·57-s − 1.04·59-s + 0.512·61-s − 0.503·63-s − 1.71·67-s − 2.40·69-s − 2.10·73-s + 1.80·79-s + 11/9·81-s − 1.09·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 640000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 640000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8854479568\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8854479568\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 10 T + 50 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 14 T + 98 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 14 T + 98 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 106 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 18 T + 162 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 10 T + 50 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
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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
−10.47323284238860517513303746085, −10.42727915056114329256195538698, −9.488524840971936076083973908355, −9.257745077761281297511885647867, −8.889944729405941387583508447683, −8.553290540490639525526282127467, −7.60988823701758753306711609889, −7.51017789456676511341077021948, −6.97076214023336553651495945209, −6.28792187574644917176663627533, −6.12675933683217027754530446182, −5.89153227177195551737873166954, −5.17186188785891799092557431681, −4.63393128301046907180296429534, −4.10317238662504806635544145188, −3.84473988813559208951878659899, −2.70443063703128147632660438434, −2.56472738755344065869744573878, −1.36605807350370056648760456486, −0.54149883447440866091580596592,
0.54149883447440866091580596592, 1.36605807350370056648760456486, 2.56472738755344065869744573878, 2.70443063703128147632660438434, 3.84473988813559208951878659899, 4.10317238662504806635544145188, 4.63393128301046907180296429534, 5.17186188785891799092557431681, 5.89153227177195551737873166954, 6.12675933683217027754530446182, 6.28792187574644917176663627533, 6.97076214023336553651495945209, 7.51017789456676511341077021948, 7.60988823701758753306711609889, 8.553290540490639525526282127467, 8.889944729405941387583508447683, 9.257745077761281297511885647867, 9.488524840971936076083973908355, 10.42727915056114329256195538698, 10.47323284238860517513303746085
