| L(s) = 1 | + 2·3-s − 2·4-s + 2·5-s + 3·9-s + 2·11-s − 4·12-s − 4·13-s + 4·15-s − 2·17-s + 2·19-s − 4·20-s − 2·23-s + 3·25-s + 4·27-s + 6·29-s − 4·31-s + 4·33-s − 6·36-s − 4·37-s − 8·39-s − 8·41-s − 10·43-s − 4·44-s + 6·45-s − 12·47-s − 4·51-s + 8·52-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 4-s + 0.894·5-s + 9-s + 0.603·11-s − 1.15·12-s − 1.10·13-s + 1.03·15-s − 0.485·17-s + 0.458·19-s − 0.894·20-s − 0.417·23-s + 3/5·25-s + 0.769·27-s + 1.11·29-s − 0.718·31-s + 0.696·33-s − 36-s − 0.657·37-s − 1.28·39-s − 1.24·41-s − 1.52·43-s − 0.603·44-s + 0.894·45-s − 1.75·47-s − 0.560·51-s + 1.10·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 65367225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65367225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 2 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 4 T + 28 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 2 T + 27 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 2 T + 31 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 2 T + 39 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 6 T + 49 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 4 T + 64 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 4 T + 60 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 8 T + 48 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 10 T + 109 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 12 T + 112 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 6 T + 83 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 26 T + 285 T^{2} + 26 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 10 T + 115 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 71 | $D_{4}$ | \( 1 + 8 T + 108 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 74 T^{2} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 8 T + 156 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 6 T - 25 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 14 T + 177 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 6 T + 185 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
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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
−7.56174331273107078164263594046, −7.35203360270829599683993473293, −7.14065913490563051990955599117, −6.57095668838249909440886915107, −6.26094649449828361038821997882, −6.13767064604234842657765846654, −5.34272494879067402688619270275, −5.12242136128974109712200954726, −4.69719698794166872590891515164, −4.65107833752487884874732894897, −4.04789712918058908047779330322, −3.76192815822588027065191961040, −3.04601263567422687128710131536, −3.03364038815468543409924224353, −2.58671467929373741813007260320, −1.86967822854792940695584010278, −1.56461849471550957143021182576, −1.35112993256476022195046873861, 0, 0,
1.35112993256476022195046873861, 1.56461849471550957143021182576, 1.86967822854792940695584010278, 2.58671467929373741813007260320, 3.03364038815468543409924224353, 3.04601263567422687128710131536, 3.76192815822588027065191961040, 4.04789712918058908047779330322, 4.65107833752487884874732894897, 4.69719698794166872590891515164, 5.12242136128974109712200954726, 5.34272494879067402688619270275, 6.13767064604234842657765846654, 6.26094649449828361038821997882, 6.57095668838249909440886915107, 7.14065913490563051990955599117, 7.35203360270829599683993473293, 7.56174331273107078164263594046
