| L(s) = 1 | − 2·5-s + 2·7-s − 2·11-s − 6·17-s − 4·19-s − 2·23-s − 2·25-s − 12·29-s + 4·31-s − 4·35-s − 10·41-s − 8·43-s + 12·47-s + 3·49-s − 16·53-s + 4·55-s − 4·59-s − 16·67-s + 10·71-s − 12·73-s − 4·77-s + 8·79-s + 8·83-s + 12·85-s − 18·89-s + 8·95-s + 4·97-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 0.755·7-s − 0.603·11-s − 1.45·17-s − 0.917·19-s − 0.417·23-s − 2/5·25-s − 2.22·29-s + 0.718·31-s − 0.676·35-s − 1.56·41-s − 1.21·43-s + 1.75·47-s + 3/7·49-s − 2.19·53-s + 0.539·55-s − 0.520·59-s − 1.95·67-s + 1.18·71-s − 1.40·73-s − 0.455·77-s + 0.900·79-s + 0.878·83-s + 1.30·85-s − 1.90·89-s + 0.820·95-s + 0.406·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4064256 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4064256 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 + 2 T + 6 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 2 T + 18 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 6 T + 38 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 4 T + 22 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_4$ | \( 1 + 12 T + 74 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_4$ | \( 1 - 4 T + 46 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 54 T^{2} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 10 T + 102 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 8 T + 22 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 12 T + 110 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 59 | $D_{4}$ | \( 1 + 4 T + 102 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 102 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 71 | $D_{4}$ | \( 1 - 10 T + 162 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 12 T + 102 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 8 T + 94 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 8 T + 102 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 18 T + 214 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 4 T + 118 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
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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
−8.816664670579689072205446465984, −8.524140477592536583575929233463, −8.044935415964780456599891417684, −7.989514912789450967345221964783, −7.34588947591534583179457565043, −7.21176453331362255378164267915, −6.55385952227453069410061744256, −6.28927536112790980543567555825, −5.69873017304813504245223522594, −5.35150540087833672858261124945, −4.71395853938282258726457258183, −4.53227153719972608956122930645, −4.03625082992940919336306418432, −3.65215265582950523097309057002, −3.07577445742729875512503092582, −2.40887772327445506884430600239, −1.92495475439801673211672945124, −1.47524465119004406724259792811, 0, 0,
1.47524465119004406724259792811, 1.92495475439801673211672945124, 2.40887772327445506884430600239, 3.07577445742729875512503092582, 3.65215265582950523097309057002, 4.03625082992940919336306418432, 4.53227153719972608956122930645, 4.71395853938282258726457258183, 5.35150540087833672858261124945, 5.69873017304813504245223522594, 6.28927536112790980543567555825, 6.55385952227453069410061744256, 7.21176453331362255378164267915, 7.34588947591534583179457565043, 7.989514912789450967345221964783, 8.044935415964780456599891417684, 8.524140477592536583575929233463, 8.816664670579689072205446465984
