| L(s) = 1 | − 3·4-s + 6·7-s + 14·13-s + 4·16-s + 4·19-s − 18·28-s + 12·31-s + 10·37-s + 14·43-s + 5·49-s − 42·52-s − 2·61-s − 9·64-s + 24·67-s + 20·73-s − 12·76-s − 24·79-s + 84·91-s + 16·97-s + 22·103-s + 2·109-s + 24·112-s − 39·121-s − 36·124-s + 127-s + 131-s + 24·133-s + ⋯ |
| L(s) = 1 | − 3/2·4-s + 2.26·7-s + 3.88·13-s + 16-s + 0.917·19-s − 3.40·28-s + 2.15·31-s + 1.64·37-s + 2.13·43-s + 5/7·49-s − 5.82·52-s − 0.256·61-s − 9/8·64-s + 2.93·67-s + 2.34·73-s − 1.37·76-s − 2.70·79-s + 8.80·91-s + 1.62·97-s + 2.16·103-s + 0.191·109-s + 2.26·112-s − 3.54·121-s − 3.23·124-s + 0.0887·127-s + 0.0873·131-s + 2.08·133-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(8.518802073\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.518802073\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| good | 2 | $C_2^3$ | \( 1 + 3 T^{2} + 5 T^{4} + 3 p^{2} T^{6} + p^{4} T^{8} \) |
| 7 | $D_{4}$ | \( ( 1 - 3 T + 11 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 11 | $D_4\times C_2$ | \( 1 + 39 T^{2} + 617 T^{4} + 39 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 - 7 T + 33 T^{2} - 7 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 - 21 T^{2} + 557 T^{4} - 21 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 - 2 T + 18 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 + p T^{2} + 9 T^{4} + p^{3} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 + 30 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{4} \) |
| 37 | $D_{4}$ | \( ( 1 - 5 T + 75 T^{2} - 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 + 39 T^{2} + 3317 T^{4} + 39 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 - 7 T + 51 T^{2} - 7 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 + 150 T^{2} + 9707 T^{4} + 150 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 45 T^{2} + 5237 T^{4} - 45 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 + 51 T^{2} + 6977 T^{4} + 51 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 + T + 75 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_{4}$ | \( ( 1 - 12 T + 149 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $D_4\times C_2$ | \( 1 + 54 T^{2} + 9467 T^{4} + 54 p^{2} T^{6} + p^{4} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 - 10 T + 87 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 + 12 T + 110 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + 139 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 + 176 T^{2} + 16782 T^{4} + 176 p^{2} T^{6} + p^{4} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 - 8 T + 126 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.50313163483949715489408340821, −6.08544828959621989346760450056, −5.98756582722258011307975137505, −5.92150531108931243766609517829, −5.79751600310262488632957488625, −5.32633861226298195836424151275, −5.18628450423962174774927548193, −4.90346258810510879206454211901, −4.84871354926068446889526822982, −4.47109302367516496092323197884, −4.43167461754631413956279969384, −4.18441333786860286937544597568, −3.99425945140642243752082070908, −3.62907345468926806915947181898, −3.52860954545473017983365908833, −3.34000397571431581182072193640, −3.01997452579702250097218453902, −2.48369088098561818909828567514, −2.44153497502643673858777721562, −2.02773436906638763983286891727, −1.50224796298394606716622812936, −1.38738661403581394858376937450, −1.12901419474777228902894123793, −0.74820769620462959198799314991, −0.70674333408368792417667107055,
0.70674333408368792417667107055, 0.74820769620462959198799314991, 1.12901419474777228902894123793, 1.38738661403581394858376937450, 1.50224796298394606716622812936, 2.02773436906638763983286891727, 2.44153497502643673858777721562, 2.48369088098561818909828567514, 3.01997452579702250097218453902, 3.34000397571431581182072193640, 3.52860954545473017983365908833, 3.62907345468926806915947181898, 3.99425945140642243752082070908, 4.18441333786860286937544597568, 4.43167461754631413956279969384, 4.47109302367516496092323197884, 4.84871354926068446889526822982, 4.90346258810510879206454211901, 5.18628450423962174774927548193, 5.32633861226298195836424151275, 5.79751600310262488632957488625, 5.92150531108931243766609517829, 5.98756582722258011307975137505, 6.08544828959621989346760450056, 6.50313163483949715489408340821
