| L(s) = 1 | + 4·3-s + 2·5-s + 4·7-s + 10·9-s + 3·11-s + 4·13-s + 8·15-s + 3·17-s + 4·19-s + 16·21-s + 8·23-s − 25-s + 20·27-s + 4·29-s − 9·31-s + 12·33-s + 8·35-s + 3·37-s + 16·39-s + 6·41-s + 8·43-s + 20·45-s − 15·47-s + 10·49-s + 12·51-s + 13·53-s + 6·55-s + ⋯ |
| L(s) = 1 | + 2.30·3-s + 0.894·5-s + 1.51·7-s + 10/3·9-s + 0.904·11-s + 1.10·13-s + 2.06·15-s + 0.727·17-s + 0.917·19-s + 3.49·21-s + 1.66·23-s − 1/5·25-s + 3.84·27-s + 0.742·29-s − 1.61·31-s + 2.08·33-s + 1.35·35-s + 0.493·37-s + 2.56·39-s + 0.937·41-s + 1.21·43-s + 2.98·45-s − 2.18·47-s + 10/7·49-s + 1.68·51-s + 1.78·53-s + 0.809·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 7^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 7^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(60.64675166\) |
| \(L(\frac12)\) |
\(\approx\) |
\(60.64675166\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{4} \) |
| 7 | $C_1$ | \( ( 1 - T )^{4} \) |
| 13 | $C_1$ | \( ( 1 - T )^{4} \) |
| good | 5 | $C_2 \wr S_4$ | \( 1 - 2 T + p T^{2} - 6 T^{3} + 4 T^{4} - 6 p T^{5} + p^{3} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $C_2 \wr S_4$ | \( 1 - 3 T + 10 T^{2} + 13 T^{3} + 26 T^{4} + 13 p T^{5} + 10 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2 \wr S_4$ | \( 1 - 3 T + 32 T^{2} - 133 T^{3} + 526 T^{4} - 133 p T^{5} + 32 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_2 \wr S_4$ | \( 1 - 4 T + 31 T^{2} - 68 T^{3} + 440 T^{4} - 68 p T^{5} + 31 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $C_2 \wr S_4$ | \( 1 - 8 T + 51 T^{2} - 320 T^{3} + 2040 T^{4} - 320 p T^{5} + 51 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $C_2 \wr S_4$ | \( 1 - 4 T + 71 T^{2} - 312 T^{3} + 2544 T^{4} - 312 p T^{5} + 71 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_2 \wr S_4$ | \( 1 + 9 T + 92 T^{2} + 613 T^{3} + 3526 T^{4} + 613 p T^{5} + 92 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_2 \wr S_4$ | \( 1 - 3 T + 112 T^{2} - 313 T^{3} + 5566 T^{4} - 313 p T^{5} + 112 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2 \wr S_4$ | \( 1 - 6 T + 136 T^{2} - 602 T^{3} + 7854 T^{4} - 602 p T^{5} + 136 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $C_2 \wr S_4$ | \( 1 - 8 T + 131 T^{2} - 800 T^{3} + 8320 T^{4} - 800 p T^{5} + 131 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $C_2 \wr S_4$ | \( 1 + 15 T + 232 T^{2} + 1931 T^{3} + 16622 T^{4} + 1931 p T^{5} + 232 p^{2} T^{6} + 15 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2 \wr S_4$ | \( 1 - 13 T + 70 T^{2} - 463 T^{3} + 4946 T^{4} - 463 p T^{5} + 70 p^{2} T^{6} - 13 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_2 \wr S_4$ | \( 1 + 8 T + 148 T^{2} + 1064 T^{3} + 12806 T^{4} + 1064 p T^{5} + 148 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2 \wr S_4$ | \( 1 - 9 T + 14 T^{2} + 197 T^{3} + 322 T^{4} + 197 p T^{5} + 14 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $C_2 \wr S_4$ | \( 1 + 8 T^{2} + 32 T^{3} + 8158 T^{4} + 32 p T^{5} + 8 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $C_2 \wr S_4$ | \( 1 - 8 T + 196 T^{2} - 1352 T^{3} + 20054 T^{4} - 1352 p T^{5} + 196 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2 \wr S_4$ | \( 1 - 32 T + 611 T^{2} - 8012 T^{3} + 79456 T^{4} - 8012 p T^{5} + 611 p^{2} T^{6} - 32 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_2 \wr S_4$ | \( 1 - T + 198 T^{2} + 387 T^{3} + 17970 T^{4} + 387 p T^{5} + 198 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_2 \wr S_4$ | \( 1 + 13 T + 254 T^{2} + 2165 T^{3} + 28530 T^{4} + 2165 p T^{5} + 254 p^{2} T^{6} + 13 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $C_2 \wr S_4$ | \( 1 - 7 T + 334 T^{2} - 1657 T^{3} + 43282 T^{4} - 1657 p T^{5} + 334 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $C_2 \wr S_4$ | \( 1 - 19 T + 318 T^{2} - 4413 T^{3} + 47202 T^{4} - 4413 p T^{5} + 318 p^{2} T^{6} - 19 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−5.90422258242948801682912218401, −5.60139204724817769209709674723, −5.52171666837116827739117514921, −5.25739083034583586050401549370, −5.14678292748948662956639622429, −4.80427841482444002098961207124, −4.64475999839476868617412729710, −4.53613160882751697037217117831, −4.44965224124263669392407818930, −3.76225351956494245566245814105, −3.73406822142759606584772722213, −3.68617090587645450551386620061, −3.64798669564500641208541952217, −3.27410385322269359602672137151, −3.05824327428536935663769153341, −2.74071002283353410936321475172, −2.56627246969851508750828492399, −2.30490537793436133212303627646, −1.92104806739094324493716845180, −1.90397859870249843010646188790, −1.83344821327538226651826415780, −1.31628656141869859854913556079, −0.967781568455477424527698329537, −0.911969848488873060481422557518, −0.77697084901374405030250800983,
0.77697084901374405030250800983, 0.911969848488873060481422557518, 0.967781568455477424527698329537, 1.31628656141869859854913556079, 1.83344821327538226651826415780, 1.90397859870249843010646188790, 1.92104806739094324493716845180, 2.30490537793436133212303627646, 2.56627246969851508750828492399, 2.74071002283353410936321475172, 3.05824327428536935663769153341, 3.27410385322269359602672137151, 3.64798669564500641208541952217, 3.68617090587645450551386620061, 3.73406822142759606584772722213, 3.76225351956494245566245814105, 4.44965224124263669392407818930, 4.53613160882751697037217117831, 4.64475999839476868617412729710, 4.80427841482444002098961207124, 5.14678292748948662956639622429, 5.25739083034583586050401549370, 5.52171666837116827739117514921, 5.60139204724817769209709674723, 5.90422258242948801682912218401
