L(s) = 1 | + 4·3-s − 2·4-s + 10·9-s − 8·12-s + 3·16-s + 8·17-s − 2·25-s + 20·27-s − 24·29-s − 20·36-s − 16·43-s + 12·48-s + 12·49-s + 32·51-s + 8·53-s + 24·61-s − 4·64-s − 16·68-s − 8·75-s + 32·79-s + 35·81-s − 96·87-s + 4·100-s − 8·101-s − 16·103-s − 16·107-s − 40·108-s + ⋯ |
L(s) = 1 | + 2.30·3-s − 4-s + 10/3·9-s − 2.30·12-s + 3/4·16-s + 1.94·17-s − 2/5·25-s + 3.84·27-s − 4.45·29-s − 3.33·36-s − 2.43·43-s + 1.73·48-s + 12/7·49-s + 4.48·51-s + 1.09·53-s + 3.07·61-s − 1/2·64-s − 1.94·68-s − 0.923·75-s + 3.60·79-s + 35/9·81-s − 10.2·87-s + 2/5·100-s − 0.796·101-s − 1.57·103-s − 1.54·107-s − 3.84·108-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(9.675745195\) |
\(L(\frac12)\) |
\(\approx\) |
\(9.675745195\) |
\(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 3 | $C_1$ | \( ( 1 - T )^{4} \) |
| 5 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 13 | | \( 1 \) |
good | 7 | $C_2^2$ | \( ( 1 - 6 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $D_{4}$ | \( ( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 - 30 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 26 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $D_{4}$ | \( ( 1 + 12 T + 86 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - 46 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 12 T^{2} - 1834 T^{4} - 12 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $D_4\times C_2$ | \( 1 - 92 T^{2} + 4966 T^{4} - 92 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 + 8 T + 70 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 30 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $D_{4}$ | \( ( 1 - 4 T - 18 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 - 44 T^{2} - 746 T^{4} - 44 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 - 102 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 110 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 76 T^{2} + 1734 T^{4} - 76 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 - 16 T + 190 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 + 20 T^{2} - 4554 T^{4} + 20 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 92 T^{2} + 5158 T^{4} - 92 p^{2} T^{6} + p^{4} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 - 300 T^{2} + 40166 T^{4} - 300 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−5.71396093458859458436070845236, −5.54048556802976233128648512309, −5.44960185031889371701694748888, −5.25254920105651099885189514911, −5.03224949295865613549263309949, −4.72411912330176563246726786870, −4.71006263232220458835281489748, −4.15894528739042375936107893188, −4.14795825929209062458366698872, −3.90345279784937682521860374131, −3.76449737491834952027846304472, −3.47858023847741537526223747609, −3.44659220496184176380521968067, −3.41429172174849349860827423339, −3.23334302545409057825073869560, −2.54403641019885675642643411216, −2.43028423342152837668000240245, −2.32477018535824715137719875716, −2.30945374016526034807096000167, −1.67748287910848453309234054537, −1.64352667013730545929929137312, −1.25154585277326469834561750855, −1.20457558474756277933388759389, −0.48404418645456168450557599645, −0.41943137717882512328225123036,
0.41943137717882512328225123036, 0.48404418645456168450557599645, 1.20457558474756277933388759389, 1.25154585277326469834561750855, 1.64352667013730545929929137312, 1.67748287910848453309234054537, 2.30945374016526034807096000167, 2.32477018535824715137719875716, 2.43028423342152837668000240245, 2.54403641019885675642643411216, 3.23334302545409057825073869560, 3.41429172174849349860827423339, 3.44659220496184176380521968067, 3.47858023847741537526223747609, 3.76449737491834952027846304472, 3.90345279784937682521860374131, 4.14795825929209062458366698872, 4.15894528739042375936107893188, 4.71006263232220458835281489748, 4.72411912330176563246726786870, 5.03224949295865613549263309949, 5.25254920105651099885189514911, 5.44960185031889371701694748888, 5.54048556802976233128648512309, 5.71396093458859458436070845236