| L(s) = 1 | + 2·3-s + 2·5-s + 2·7-s + 2·9-s − 2·13-s + 4·15-s + 2·17-s − 8·19-s + 4·21-s − 10·23-s − 25-s + 6·27-s + 4·35-s − 10·37-s − 4·39-s + 12·41-s − 6·43-s + 4·45-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 − 4·65-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 0.894·5-s + 0.755·7-s + 2/3·9-s − 0.554·13-s + 1.03·15-s + 0.485·17-s − 1.83·19-s + 0.872·21-s − 2.08·23-s − 1/5·25-s + 1.15·27-s + 0.676·35-s − 1.64·37-s − 0.640·39-s + 1.87·41-s − 0.914·43-s + 0.596·45-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 − 0.496·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.992257902\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.992257902\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| 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 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 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
−13.25228226055873729293862743578, −12.61011343313657734466065240495, −12.34789578347853819677779384122, −11.76159118974839900845605201616, −10.95012515232120247188036202485, −10.59254263417348584777203866687, −9.887104930470897720256197454554, −9.737530630017934070074751495687, −8.994786825222327311658351953573, −8.484163872031317758065800093479, −7.895872749771244259331481739921, −7.84145952511783756878884078215, −6.52982603632530620059916845267, −6.48592670710975906264796631890, −5.46933778201520942345358131072, −4.85387390315498983689927577944, −4.10282828350672403813017716725, −3.36002871228799822622667393158, −2.14563453935508534164070954675, −2.00506365811512203597125664732,
2.00506365811512203597125664732, 2.14563453935508534164070954675, 3.36002871228799822622667393158, 4.10282828350672403813017716725, 4.85387390315498983689927577944, 5.46933778201520942345358131072, 6.48592670710975906264796631890, 6.52982603632530620059916845267, 7.84145952511783756878884078215, 7.895872749771244259331481739921, 8.484163872031317758065800093479, 8.994786825222327311658351953573, 9.737530630017934070074751495687, 9.887104930470897720256197454554, 10.59254263417348584777203866687, 10.95012515232120247188036202485, 11.76159118974839900845605201616, 12.34789578347853819677779384122, 12.61011343313657734466065240495, 13.25228226055873729293862743578
