| L(s) = 1 | − 2·3-s − 2·5-s + 4·7-s − 9-s − 2·11-s − 2·13-s + 4·15-s − 4·19-s − 8·21-s + 8·23-s − 7·25-s + 6·27-s − 2·29-s + 2·31-s + 4·33-s − 8·35-s + 4·39-s − 4·41-s − 2·43-s + 2·45-s + 6·47-s − 2·49-s − 6·53-s + 4·55-s + 8·57-s − 8·59-s − 8·61-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 0.894·5-s + 1.51·7-s − 1/3·9-s − 0.603·11-s − 0.554·13-s + 1.03·15-s − 0.917·19-s − 1.74·21-s + 1.66·23-s − 7/5·25-s + 1.15·27-s − 0.371·29-s + 0.359·31-s + 0.696·33-s − 1.35·35-s + 0.640·39-s − 0.624·41-s − 0.304·43-s + 0.298·45-s + 0.875·47-s − 2/7·49-s − 0.824·53-s + 0.539·55-s + 1.05·57-s − 1.04·59-s − 1.02·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3444736 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3444736 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 29 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 + 2 T + 5 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 11 | $D_{4}$ | \( 1 + 2 T + 21 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 2 T + 19 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 4 T + 10 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 8 T + 54 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 2 T + 61 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 42 T^{2} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 4 T + 78 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 2 T + 69 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 6 T + 101 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 6 T + 43 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 8 T + 62 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 8 T + 66 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_4$ | \( 1 - 12 T + 170 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 79 | $D_{4}$ | \( 1 - 2 T + 61 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 8 T + 110 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 12 T + 206 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 12 T + 222 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
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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
−8.901960975109664795419360457665, −8.537413695594600796247411351183, −8.223120684904827822914433777598, −7.84034813238954427733937353300, −7.52289838239714853495968658596, −7.22502405939388722231513331229, −6.44257538556520353211922172701, −6.38687301958361764683147926388, −5.60318585034145658692382574125, −5.44057229537643532192526269848, −4.92371286068695655605453713601, −4.73643178173562166120859781892, −4.16986857321832827512211503839, −3.79202341668985581826466242986, −2.82110046633056604146741916747, −2.76786087527848586964829931745, −1.74583122288390127477150012427, −1.32852997150164032090909125122, 0, 0,
1.32852997150164032090909125122, 1.74583122288390127477150012427, 2.76786087527848586964829931745, 2.82110046633056604146741916747, 3.79202341668985581826466242986, 4.16986857321832827512211503839, 4.73643178173562166120859781892, 4.92371286068695655605453713601, 5.44057229537643532192526269848, 5.60318585034145658692382574125, 6.38687301958361764683147926388, 6.44257538556520353211922172701, 7.22502405939388722231513331229, 7.52289838239714853495968658596, 7.84034813238954427733937353300, 8.223120684904827822914433777598, 8.537413695594600796247411351183, 8.901960975109664795419360457665
