| L(s) = 1 | − 4·3-s − 4·5-s − 4·7-s + 6·9-s − 4·13-s + 16·15-s − 4·17-s + 16·21-s − 12·23-s + 2·25-s + 4·27-s − 8·29-s + 16·35-s + 12·37-s + 16·39-s − 4·43-s − 24·45-s − 20·47-s + 8·49-s + 16·51-s − 12·53-s − 16·59-s − 24·61-s − 24·63-s + 16·65-s − 12·67-s + 48·69-s + ⋯ |
| L(s) = 1 | − 2.30·3-s − 1.78·5-s − 1.51·7-s + 2·9-s − 1.10·13-s + 4.13·15-s − 0.970·17-s + 3.49·21-s − 2.50·23-s + 2/5·25-s + 0.769·27-s − 1.48·29-s + 2.70·35-s + 1.97·37-s + 2.56·39-s − 0.609·43-s − 3.57·45-s − 2.91·47-s + 8/7·49-s + 2.24·51-s − 1.64·53-s − 2.08·59-s − 3.07·61-s − 3.02·63-s + 1.98·65-s − 1.46·67-s + 5.77·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{4} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{4} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
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 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| good | 7 | $D_4\times C_2$ | \( 1 + 4 T + 8 T^{2} + 20 T^{3} + 46 T^{4} + 20 p T^{5} + 8 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 20 T^{2} + 214 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $D_4\times C_2$ | \( 1 + 4 T + 8 T^{2} - 4 T^{3} - 194 T^{4} - 4 p T^{5} + 8 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 + 4 T + 8 T^{2} + 12 T^{3} - 178 T^{4} + 12 p T^{5} + 8 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 - 52 T^{2} + 1270 T^{4} - 52 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 + 12 T + 72 T^{2} + 444 T^{3} + 2542 T^{4} + 444 p T^{5} + 72 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 + 4 T + 30 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + 30 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 12 T + 72 T^{2} - 468 T^{3} + 3038 T^{4} - 468 p T^{5} + 72 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2^2$ | \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 + 4 T + 8 T^{2} + 164 T^{3} + 3358 T^{4} + 164 p T^{5} + 8 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 + 20 T + 200 T^{2} + 1860 T^{3} + 15182 T^{4} + 1860 p T^{5} + 200 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2^2$ | \( ( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 61 | $D_{4}$ | \( ( 1 + 12 T + 126 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 + 12 T + 72 T^{2} - 180 T^{3} - 6274 T^{4} - 180 p T^{5} + 72 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 - 132 T^{2} + 13286 T^{4} - 132 p^{2} T^{6} + p^{4} T^{8} \) |
| 73 | $D_4\times C_2$ | \( 1 - 4 T + 8 T^{2} - 44 T^{3} - 3602 T^{4} - 44 p T^{5} + 8 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 - 292 T^{2} + 33670 T^{4} - 292 p^{2} T^{6} + p^{4} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 + 4 T + 8 T^{2} + 196 T^{3} + 3646 T^{4} + 196 p T^{5} + 8 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 + 20 T + 246 T^{2} + 20 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.69717823286276099637863263242, −7.20511648754927071640107524130, −7.09861619378050080189685092234, −6.96169799701934089203530988463, −6.79419869263108511277147332578, −6.20757758826061290540124485319, −6.18598394200461733688175091936, −6.14004799815225285207141787211, −6.04517976364497022416598265293, −5.64612523466996702002035902365, −5.47947161797779997019372768288, −5.05597908403725179781238658724, −4.88862071236878151754590629911, −4.46746354144859154210867854418, −4.45532033563250869191455811760, −4.24736842111125580443402211540, −4.05646868522523180975860633220, −3.60237164014713376360508706984, −3.35172340300025719156540590900, −3.04848167123032299079881373222, −2.76671104378722245640245172720, −2.64551494248652101268061778797, −1.83523356118171757667998848040, −1.70408717219786167580353051772, −1.20398726023577168611881348658, 0, 0, 0, 0,
1.20398726023577168611881348658, 1.70408717219786167580353051772, 1.83523356118171757667998848040, 2.64551494248652101268061778797, 2.76671104378722245640245172720, 3.04848167123032299079881373222, 3.35172340300025719156540590900, 3.60237164014713376360508706984, 4.05646868522523180975860633220, 4.24736842111125580443402211540, 4.45532033563250869191455811760, 4.46746354144859154210867854418, 4.88862071236878151754590629911, 5.05597908403725179781238658724, 5.47947161797779997019372768288, 5.64612523466996702002035902365, 6.04517976364497022416598265293, 6.14004799815225285207141787211, 6.18598394200461733688175091936, 6.20757758826061290540124485319, 6.79419869263108511277147332578, 6.96169799701934089203530988463, 7.09861619378050080189685092234, 7.20511648754927071640107524130, 7.69717823286276099637863263242
