| L(s) = 1 | − 2·3-s − 8·7-s + 3·9-s − 12·13-s − 12·17-s + 2·19-s + 16·21-s − 8·25-s − 10·27-s + 24·39-s − 6·41-s − 8·43-s − 12·47-s + 24·49-s + 24·51-s − 12·53-s − 4·57-s − 18·59-s + 8·61-s − 24·63-s + 6·67-s − 12·71-s + 2·73-s + 16·75-s + 20·81-s + 24·89-s + 96·91-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 3.02·7-s + 9-s − 3.32·13-s − 2.91·17-s + 0.458·19-s + 3.49·21-s − 8/5·25-s − 1.92·27-s + 3.84·39-s − 0.937·41-s − 1.21·43-s − 1.75·47-s + 24/7·49-s + 3.36·51-s − 1.64·53-s − 0.529·57-s − 2.34·59-s + 1.02·61-s − 3.02·63-s + 0.733·67-s − 1.42·71-s + 0.234·73-s + 1.84·75-s + 20/9·81-s + 2.54·89-s + 10.0·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.01193698164\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.01193698164\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 + 2 T + T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 2 T - 15 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| good | 5 | $C_2^3$ | \( 1 + 8 T^{2} + 39 T^{4} + 8 p^{2} T^{6} + p^{4} T^{8} \) |
| 7 | $D_{4}$ | \( ( 1 + 4 T + 12 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 11 | $D_4\times C_2$ | \( 1 - 34 T^{2} + 507 T^{4} - 34 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $C_2^2$ | \( ( 1 + 6 T + 25 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 + 12 T + 86 T^{2} + 456 T^{3} + 1971 T^{4} + 456 p T^{5} + 86 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $C_2^3$ | \( 1 - 4 T^{2} - 513 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2^3$ | \( 1 - 52 T^{2} + 1863 T^{4} - 52 p^{2} T^{6} + p^{4} T^{8} \) |
| 31 | $C_2^2$ | \( ( 1 - 44 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 88 T^{2} + 3810 T^{4} - 88 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $C_2^2$ | \( ( 1 + 3 T - 32 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 + 8 T - 14 T^{2} - 64 T^{3} + 1483 T^{4} - 64 p T^{5} - 14 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 + 12 T + 56 T^{2} + 96 T^{3} + 111 T^{4} + 96 p T^{5} + 56 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 + 12 T + 26 T^{2} + 144 T^{3} + 3483 T^{4} + 144 p T^{5} + 26 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 + 18 T + 131 T^{2} + 1350 T^{3} + 14652 T^{4} + 1350 p T^{5} + 131 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 8 T - 68 T^{2} - 80 T^{3} + 9799 T^{4} - 80 p T^{5} - 68 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 6 T + 131 T^{2} - 714 T^{3} + 10476 T^{4} - 714 p T^{5} + 131 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2^2$ | \( ( 1 + 6 T - 35 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 2 T - 47 T^{2} + 190 T^{3} - 3020 T^{4} + 190 p T^{5} - 47 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_2^3$ | \( 1 + 86 T^{2} + 1155 T^{4} + 86 p^{2} T^{6} + p^{4} T^{8} \) |
| 83 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 18 T + 241 T^{2} - 18 p T^{3} + p^{2} T^{4} )( 1 + 18 T + 241 T^{2} + 18 p T^{3} + p^{2} T^{4} ) \) |
| 89 | $D_4\times C_2$ | \( 1 - 24 T + 278 T^{2} - 2880 T^{3} + 29619 T^{4} - 2880 p T^{5} + 278 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 + 18 T + 257 T^{2} + 2682 T^{3} + 23268 T^{4} + 2682 p T^{5} + 257 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−7.05780558869616356742417207119, −6.99511427734581641421438442056, −6.78435024665851196463523856237, −6.42589831402705867431444599857, −6.40990099497161945904219905037, −6.22879404047453459279853661990, −6.08373015508812769471420596752, −5.63597412495698544717602565187, −5.29630988239453124757859994076, −5.20968726745541476963587163567, −4.94889830160876964978098272821, −4.64705686724673619814543687516, −4.51686359058294171846129145716, −4.14948603731631495642795911622, −4.02016800263063226700447286197, −3.49723176188910547344448924852, −3.25036604926584370314173885763, −3.11348767342970864004135789721, −2.86463956887947738533774468744, −2.24369966125123557714220916792, −2.17816461713731591660597698999, −1.96322193879449765741878637298, −1.39882601037000971575206727618, −0.20206774888494465385955466473, −0.12092908824677507958414350416,
0.12092908824677507958414350416, 0.20206774888494465385955466473, 1.39882601037000971575206727618, 1.96322193879449765741878637298, 2.17816461713731591660597698999, 2.24369966125123557714220916792, 2.86463956887947738533774468744, 3.11348767342970864004135789721, 3.25036604926584370314173885763, 3.49723176188910547344448924852, 4.02016800263063226700447286197, 4.14948603731631495642795911622, 4.51686359058294171846129145716, 4.64705686724673619814543687516, 4.94889830160876964978098272821, 5.20968726745541476963587163567, 5.29630988239453124757859994076, 5.63597412495698544717602565187, 6.08373015508812769471420596752, 6.22879404047453459279853661990, 6.40990099497161945904219905037, 6.42589831402705867431444599857, 6.78435024665851196463523856237, 6.99511427734581641421438442056, 7.05780558869616356742417207119
