| L(s) = 1 | − 2·4-s + 2·5-s − 10·7-s + 6·11-s − 12·13-s + 3·16-s + 6·19-s − 4·20-s + 12·23-s + 25-s + 20·28-s + 12·29-s − 20·35-s − 2·37-s − 18·41-s + 4·43-s − 12·44-s + 36·47-s + 61·49-s + 24·52-s + 6·53-s + 12·55-s + 12·59-s − 4·64-s − 24·65-s − 16·67-s + 24·73-s + ⋯ |
| L(s) = 1 | − 4-s + 0.894·5-s − 3.77·7-s + 1.80·11-s − 3.32·13-s + 3/4·16-s + 1.37·19-s − 0.894·20-s + 2.50·23-s + 1/5·25-s + 3.77·28-s + 2.22·29-s − 3.38·35-s − 0.328·37-s − 2.81·41-s + 0.609·43-s − 1.80·44-s + 5.25·47-s + 61/7·49-s + 3.32·52-s + 0.824·53-s + 1.61·55-s + 1.56·59-s − 1/2·64-s − 2.97·65-s − 1.95·67-s + 2.80·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{12} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{12} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.317354106\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.317354106\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| good | 11 | $D_4\times C_2$ | \( 1 - 6 T + 28 T^{2} - 96 T^{3} + 267 T^{4} - 96 p T^{5} + 28 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $C_2^2$ | \( ( 1 + 6 T + 25 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $D_4\times C_2$ | \( 1 - 6 T + 44 T^{2} - 192 T^{3} + 891 T^{4} - 192 p T^{5} + 44 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 12 T + 70 T^{2} - 264 T^{3} + 1059 T^{4} - 264 p T^{5} + 70 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 12 T + 109 T^{2} - 732 T^{3} + 4272 T^{4} - 732 p T^{5} + 109 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 - 52 T^{2} + 1626 T^{4} - 52 p^{2} T^{6} + p^{4} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 + 2 T - 44 T^{2} - 52 T^{3} + 787 T^{4} - 52 p T^{5} - 44 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2^2$ | \( ( 1 + 9 T + 40 T^{2} + 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 4 T - 47 T^{2} + 92 T^{3} + 1432 T^{4} + 92 p T^{5} - 47 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{4} \) |
| 53 | $D_4\times C_2$ | \( 1 - 6 T + 40 T^{2} - 168 T^{3} - 1389 T^{4} - 168 p T^{5} + 40 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 - 6 T + 100 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $D_4\times C_2$ | \( 1 - 76 T^{2} + 1974 T^{4} - 76 p^{2} T^{6} + p^{4} T^{8} \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 71 | $D_4\times C_2$ | \( 1 - 260 T^{2} + 26874 T^{4} - 260 p^{2} T^{6} + p^{4} T^{8} \) |
| 73 | $C_2^2$ | \( ( 1 - 12 T + 121 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 + 2 T - 84 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 12 T - 31 T^{2} - 108 T^{3} + 11784 T^{4} - 108 p T^{5} - 31 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 12 T + 38 T^{2} + 864 T^{3} - 9501 T^{4} + 864 p T^{5} + 38 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 - 12 T + 110 T^{2} - 744 T^{3} - 909 T^{4} - 744 p T^{5} + 110 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−6.54885975180225964199365065915, −6.32540099361653749113579248772, −6.18658118605647227081700266228, −6.15766202649455119950316445427, −5.60552767511426858886992691658, −5.43947836695903838126905788550, −5.22655983525522209762872603755, −5.13495876273666922331831676082, −5.09246341591032031801830225010, −4.56612167389823984615566539557, −4.22677302245145317175357339668, −4.20919593878625370463758896049, −4.03831601102192597669916169328, −3.41155207815732149992847154796, −3.36173934685030214848735405144, −3.32361458335404933226575743645, −3.02180239365171781950239214625, −2.67351343684557446975237215044, −2.48744062138014376242163539210, −2.26075141929556530697626330247, −2.06791263317834630561416975604, −1.22353968380630205013401019225, −0.818190669165145295128981974370, −0.75600560713121918022624579918, −0.43739839717500233483877517817,
0.43739839717500233483877517817, 0.75600560713121918022624579918, 0.818190669165145295128981974370, 1.22353968380630205013401019225, 2.06791263317834630561416975604, 2.26075141929556530697626330247, 2.48744062138014376242163539210, 2.67351343684557446975237215044, 3.02180239365171781950239214625, 3.32361458335404933226575743645, 3.36173934685030214848735405144, 3.41155207815732149992847154796, 4.03831601102192597669916169328, 4.20919593878625370463758896049, 4.22677302245145317175357339668, 4.56612167389823984615566539557, 5.09246341591032031801830225010, 5.13495876273666922331831676082, 5.22655983525522209762872603755, 5.43947836695903838126905788550, 5.60552767511426858886992691658, 6.15766202649455119950316445427, 6.18658118605647227081700266228, 6.32540099361653749113579248772, 6.54885975180225964199365065915
