| L(s) = 1 | − 4-s − 4·7-s + 2·11-s − 4·13-s − 3·16-s + 4·19-s + 4·28-s − 8·31-s − 4·37-s − 4·43-s − 2·44-s − 2·49-s + 4·52-s − 12·53-s + 4·61-s + 7·64-s − 16·67-s − 4·73-s − 4·76-s − 8·77-s − 20·79-s + 24·83-s + 12·89-s + 16·91-s + 20·97-s − 16·103-s + 24·107-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 1.51·7-s + 0.603·11-s − 1.10·13-s − 3/4·16-s + 0.917·19-s + 0.755·28-s − 1.43·31-s − 0.657·37-s − 0.609·43-s − 0.301·44-s − 2/7·49-s + 0.554·52-s − 1.64·53-s + 0.512·61-s + 7/8·64-s − 1.95·67-s − 0.468·73-s − 0.458·76-s − 0.911·77-s − 2.25·79-s + 2.63·83-s + 1.27·89-s + 1.67·91-s + 2.03·97-s − 1.57·103-s + 2.32·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6125625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6125625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 2 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 13 | $D_{4}$ | \( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $D_{4}$ | \( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 8 T + 30 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 4 T + 30 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 70 T^{2} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 4 T + 42 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 12 T + 94 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 70 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 4 T + 42 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 20 T + 246 T^{2} + 20 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 24 T + 298 T^{2} - 24 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 12 T + 166 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
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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
−8.814127321635745912038140176294, −8.607522049627000704940417341322, −7.69735998848652636947937930134, −7.66149334077370576666142539244, −7.28546833628206220907980490811, −6.60398367550682932116583418422, −6.54973818543039877891372524390, −6.21411664904222640178475119308, −5.37418924240577003230373375310, −5.37114300791228383583758529819, −4.67505474537592244856044624651, −4.46344441284911895297876364495, −3.71454119464702276352815475694, −3.44657967038959692208149858925, −3.07618491095699059708805535565, −2.48829905494733104436189766369, −1.88100372135999436373448821297, −1.21743386798162512038033609400, 0, 0,
1.21743386798162512038033609400, 1.88100372135999436373448821297, 2.48829905494733104436189766369, 3.07618491095699059708805535565, 3.44657967038959692208149858925, 3.71454119464702276352815475694, 4.46344441284911895297876364495, 4.67505474537592244856044624651, 5.37114300791228383583758529819, 5.37418924240577003230373375310, 6.21411664904222640178475119308, 6.54973818543039877891372524390, 6.60398367550682932116583418422, 7.28546833628206220907980490811, 7.66149334077370576666142539244, 7.69735998848652636947937930134, 8.607522049627000704940417341322, 8.814127321635745912038140176294
