| L(s) = 1 | − 4·2-s + 2·4-s − 3·5-s + 20·8-s + 12·10-s − 45·16-s + 8·17-s − 6·20-s + 25-s − 16·32-s − 32·34-s + 12·37-s − 60·40-s + 30·43-s + 22·47-s + 4·49-s − 4·50-s − 12·59-s + 204·64-s + 16·68-s − 24·71-s − 12·73-s − 48·74-s + 135·80-s − 24·85-s − 120·86-s − 88·94-s + ⋯ |
| L(s) = 1 | − 2.82·2-s + 4-s − 1.34·5-s + 7.07·8-s + 3.79·10-s − 11.2·16-s + 1.94·17-s − 1.34·20-s + 1/5·25-s − 2.82·32-s − 5.48·34-s + 1.97·37-s − 9.48·40-s + 4.57·43-s + 3.20·47-s + 4/7·49-s − 0.565·50-s − 1.56·59-s + 51/2·64-s + 1.94·68-s − 2.84·71-s − 1.40·73-s − 5.57·74-s + 15.0·80-s − 2.60·85-s − 12.9·86-s − 9.07·94-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{4} \cdot 29^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{4} \cdot 29^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3213810469\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3213810469\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 + 3 T + 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| good | 2 | $C_2$ | \( ( 1 + T + p T^{2} )^{4} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2}( 1 + 4 T + p T^{2} )^{2} \) |
| 11 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 9 T + 38 T^{2} - 9 p T^{3} + p^{2} T^{4} )( 1 + 9 T + 38 T^{2} + 9 p T^{3} + p^{2} T^{4} ) \) |
| 13 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - T - 12 T^{2} - p T^{3} + p^{2} T^{4} )( 1 + T - 12 T^{2} + p T^{3} + p^{2} T^{4} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2}( 1 + 8 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 - 61 T^{2} + 1896 T^{4} - 61 p^{2} T^{6} + p^{4} T^{8} \) |
| 37 | $D_{4}$ | \( ( 1 - 6 T + 66 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 - 56 T^{2} + 3534 T^{4} - 56 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 - 15 T + 138 T^{2} - 15 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $D_{4}$ | \( ( 1 - 11 T + 86 T^{2} - 11 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $D_4\times C_2$ | \( 1 - 185 T^{2} + 14136 T^{4} - 185 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 + 6 T + 110 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $D_4\times C_2$ | \( 1 - 88 T^{2} + 3870 T^{4} - 88 p^{2} T^{6} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 112 T^{2} + 6606 T^{4} - 112 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 12 T + 110 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_{4}$ | \( ( 1 + 6 T + 138 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $D_4\times C_2$ | \( 1 - 37 T^{2} + 6360 T^{4} - 37 p^{2} T^{6} + p^{4} T^{8} \) |
| 83 | $C_2^2$ | \( ( 1 - 58 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 - 248 T^{2} + 30606 T^{4} - 248 p^{2} T^{6} + p^{4} T^{8} \) |
| 97 | $C_2^2$ | \( ( 1 + 126 T^{2} + p^{2} T^{4} )^{2} \) |
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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
−7.20146714181742588133091281168, −7.08818483998618076620061008509, −6.83330253778717030876877090819, −5.99849079218762121533379562886, −5.84615666827834204040568034134, −5.82575239386262724709688265403, −5.72570744899225278627839451848, −5.48823077571278226780689652565, −4.89410338707820203713011685466, −4.75915540115618497688393689513, −4.53789034532256557920063637696, −4.29118106736854352275079961412, −4.21469540188121993168708978091, −3.97419758667111246883223223285, −3.96997391146145797869003051788, −3.30115427750447144526629581192, −3.27084912663831918723810945662, −2.80289828374888555354670818546, −2.42038869303822502622971880407, −1.89569763413408200494766835671, −1.67452713861081449236551316845, −1.03231824657616587373146160867, −0.992894084675365548830218466424, −0.61849003070582594287040890786, −0.46635058168356993466804751489,
0.46635058168356993466804751489, 0.61849003070582594287040890786, 0.992894084675365548830218466424, 1.03231824657616587373146160867, 1.67452713861081449236551316845, 1.89569763413408200494766835671, 2.42038869303822502622971880407, 2.80289828374888555354670818546, 3.27084912663831918723810945662, 3.30115427750447144526629581192, 3.96997391146145797869003051788, 3.97419758667111246883223223285, 4.21469540188121993168708978091, 4.29118106736854352275079961412, 4.53789034532256557920063637696, 4.75915540115618497688393689513, 4.89410338707820203713011685466, 5.48823077571278226780689652565, 5.72570744899225278627839451848, 5.82575239386262724709688265403, 5.84615666827834204040568034134, 5.99849079218762121533379562886, 6.83330253778717030876877090819, 7.08818483998618076620061008509, 7.20146714181742588133091281168
