| L(s) = 1 | + 2·4-s + 4·9-s − 8·11-s + 3·16-s + 8·19-s − 4·29-s − 8·31-s + 8·36-s − 24·41-s − 16·44-s + 4·49-s + 8·61-s + 12·64-s − 16·71-s + 16·76-s − 24·79-s − 6·81-s + 8·89-s − 32·99-s + 40·101-s − 8·109-s − 8·116-s + 12·121-s − 16·124-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | + 4-s + 4/3·9-s − 2.41·11-s + 3/4·16-s + 1.83·19-s − 0.742·29-s − 1.43·31-s + 4/3·36-s − 3.74·41-s − 2.41·44-s + 4/7·49-s + 1.02·61-s + 3/2·64-s − 1.89·71-s + 1.83·76-s − 2.70·79-s − 2/3·81-s + 0.847·89-s − 3.21·99-s + 3.98·101-s − 0.766·109-s − 0.742·116-s + 1.09·121-s − 1.43·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 29^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 29^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.531221073\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.531221073\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | | \( 1 \) |
| 29 | $C_1$ | \( ( 1 + T )^{4} \) |
| good | 2 | $D_4\times C_2$ | \( 1 - p T^{2} + T^{4} - p^{3} T^{6} + p^{4} T^{8} \) |
| 3 | $C_2^2$ | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 7 | $C_4\times C_2$ | \( 1 - 4 T^{2} - 26 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 26 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $D_{4}$ | \( ( 1 - 4 T + 34 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $C_4\times C_2$ | \( 1 - 4 T^{2} - 90 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} \) |
| 31 | $C_4$ | \( ( 1 + 4 T - 6 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 - 52 T^{2} + 486 T^{4} - 52 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 140 T^{2} + 10006 T^{4} - 140 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 61 | $D_{4}$ | \( ( 1 - 4 T + 94 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 116 T^{2} + 11190 T^{4} - 116 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 8 T + 30 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 74 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 + 12 T + 122 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 116 T^{2} + 13942 T^{4} - 116 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 - 4 T + 150 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 212 T^{2} + 25446 T^{4} - 212 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−7.41880192803429992872452724615, −7.25187404212121187614104464163, −7.17566378290352898151483951334, −6.98877572090141888045934799083, −6.64164854231641694923232928794, −6.21042033642033670886527893450, −6.06189807715776114358029512128, −5.95619437309432876618056402143, −5.41054621581820714585244244112, −5.31323269119523513982193402981, −5.06134966940389820443627324411, −4.99824140036229037522864058918, −4.87357471908976139441332310904, −4.09903684341191952450336855298, −4.03561866328231830489959466320, −3.78146777540943074370420425455, −3.32989835011336231261878133166, −3.06149903445071343546422001013, −3.03205754067048600552143009051, −2.42805915931343479592608498791, −2.30385375557248273607590224548, −1.82655675637076912049312911936, −1.45642858216634064691998303360, −1.28792416027748904210396706923, −0.30620756861729516670729676068,
0.30620756861729516670729676068, 1.28792416027748904210396706923, 1.45642858216634064691998303360, 1.82655675637076912049312911936, 2.30385375557248273607590224548, 2.42805915931343479592608498791, 3.03205754067048600552143009051, 3.06149903445071343546422001013, 3.32989835011336231261878133166, 3.78146777540943074370420425455, 4.03561866328231830489959466320, 4.09903684341191952450336855298, 4.87357471908976139441332310904, 4.99824140036229037522864058918, 5.06134966940389820443627324411, 5.31323269119523513982193402981, 5.41054621581820714585244244112, 5.95619437309432876618056402143, 6.06189807715776114358029512128, 6.21042033642033670886527893450, 6.64164854231641694923232928794, 6.98877572090141888045934799083, 7.17566378290352898151483951334, 7.25187404212121187614104464163, 7.41880192803429992872452724615
