| L(s) = 1 | − 8·7-s + 8·17-s + 16·23-s + 8·25-s + 24·31-s − 24·41-s − 16·47-s + 16·49-s + 48·71-s − 16·73-s + 40·79-s + 8·89-s + 8·97-s − 8·103-s − 24·113-s − 64·119-s + 36·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s − 128·161-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | − 3.02·7-s + 1.94·17-s + 3.33·23-s + 8/5·25-s + 4.31·31-s − 3.74·41-s − 2.33·47-s + 16/7·49-s + 5.69·71-s − 1.87·73-s + 4.50·79-s + 0.847·89-s + 0.812·97-s − 0.788·103-s − 2.25·113-s − 5.86·119-s + 3.27·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s − 10.0·161-s + 0.0783·163-s + 0.0773·167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.936862483\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.936862483\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| good | 5 | $D_4\times C_2$ | \( 1 - 8 T^{2} + 34 T^{4} - 8 p^{2} T^{6} + p^{4} T^{8} \) |
| 7 | $D_{4}$ | \( ( 1 + 4 T + 16 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 18 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 18 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_4$ | \( ( 1 - 4 T + 6 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 - 6 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $D_{4}$ | \( ( 1 - 8 T + 54 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $D_4\times C_2$ | \( 1 - 104 T^{2} + 4354 T^{4} - 104 p^{2} T^{6} + p^{4} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 - 12 T + 96 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 52 T^{2} + 1366 T^{4} - 52 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $C_4$ | \( ( 1 + 12 T + 86 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 76 T^{2} + 3094 T^{4} - 76 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $D_{4}$ | \( ( 1 + 8 T + 38 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $D_4\times C_2$ | \( 1 - 8 T^{2} + 4066 T^{4} - 8 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $C_2^2$ | \( ( 1 - 102 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $D_4\times C_2$ | \( 1 - 148 T^{2} + 10870 T^{4} - 148 p^{2} T^{6} + p^{4} T^{8} \) |
| 67 | $C_2^2$ | \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $D_{4}$ | \( ( 1 - 24 T + 278 T^{2} - 24 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_{4}$ | \( ( 1 + 8 T + 130 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 - 20 T + 240 T^{2} - 20 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 68 T^{2} + 12886 T^{4} - 68 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 97 | $D_{4}$ | \( ( 1 - 4 T + 70 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−5.77414396952805306337087860759, −5.70755396252070438761876379582, −5.60559142034298043320757727807, −5.10321755684354656591292197627, −4.93236063222453567819909265408, −4.90216756181858758074628981971, −4.87683826023652595636034134531, −4.60129884746033850119487031272, −4.56997907006166102498116849913, −3.72971411905691760370013194020, −3.63123233441952534398860988992, −3.58659360838161271987312636818, −3.35879468774787559909205931126, −3.28083362934196797159578034173, −3.19387178108086672220347071121, −2.81821338672261935108162561710, −2.73049184526991598124458064492, −2.40373287646076244529923128384, −2.28448613856886736564061910407, −1.75337571757815665876807099947, −1.33060982230446455789162493622, −1.10168268209762330429948364729, −0.920916341621055649172417982530, −0.71297787775766002018447504601, −0.22070097125019218705569006784,
0.22070097125019218705569006784, 0.71297787775766002018447504601, 0.920916341621055649172417982530, 1.10168268209762330429948364729, 1.33060982230446455789162493622, 1.75337571757815665876807099947, 2.28448613856886736564061910407, 2.40373287646076244529923128384, 2.73049184526991598124458064492, 2.81821338672261935108162561710, 3.19387178108086672220347071121, 3.28083362934196797159578034173, 3.35879468774787559909205931126, 3.58659360838161271987312636818, 3.63123233441952534398860988992, 3.72971411905691760370013194020, 4.56997907006166102498116849913, 4.60129884746033850119487031272, 4.87683826023652595636034134531, 4.90216756181858758074628981971, 4.93236063222453567819909265408, 5.10321755684354656591292197627, 5.60559142034298043320757727807, 5.70755396252070438761876379582, 5.77414396952805306337087860759
