| L(s) = 1 | − 3·4-s + 6·16-s + 10·17-s + 10·25-s + 20·29-s − 22·43-s + 18·49-s − 8·61-s − 10·64-s − 30·68-s − 36·79-s − 30·100-s + 12·101-s − 60·103-s − 2·107-s − 6·113-s − 60·116-s + 49·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | − 3/2·4-s + 3/2·16-s + 2.42·17-s + 2·25-s + 3.71·29-s − 3.35·43-s + 18/7·49-s − 1.02·61-s − 5/4·64-s − 3.63·68-s − 4.05·79-s − 3·100-s + 1.19·101-s − 5.91·103-s − 0.193·107-s − 0.564·113-s − 5.57·116-s + 4.45·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{12} \cdot 13^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{12} \cdot 13^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.292216134\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.292216134\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 + T^{2} )^{3} \) |
| 3 | \( 1 \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 - 2 p T^{2} + 71 T^{4} - 396 T^{6} + 71 p^{2} T^{8} - 2 p^{5} T^{10} + p^{6} T^{12} \) |
| 7 | \( 1 - 18 T^{2} + 143 T^{4} - 860 T^{6} + 143 p^{2} T^{8} - 18 p^{4} T^{10} + p^{6} T^{12} \) |
| 11 | \( 1 - 49 T^{2} + 1161 T^{4} - 16177 T^{6} + 1161 p^{2} T^{8} - 49 p^{4} T^{10} + p^{6} T^{12} \) |
| 17 | \( ( 1 - 5 T + 29 T^{2} - 73 T^{3} + 29 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 19 | \( 1 - 81 T^{2} + 3137 T^{4} - 74273 T^{6} + 3137 p^{2} T^{8} - 81 p^{4} T^{10} + p^{6} T^{12} \) |
| 23 | \( ( 1 + 41 T^{2} + 56 T^{3} + 41 p T^{4} + p^{3} T^{6} )^{2} \) |
| 29 | \( ( 1 - 10 T + 83 T^{2} - 476 T^{3} + 83 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 31 | \( 1 - 82 T^{2} + 3967 T^{4} - 137116 T^{6} + 3967 p^{2} T^{8} - 82 p^{4} T^{10} + p^{6} T^{12} \) |
| 37 | \( 1 - 138 T^{2} + 9671 T^{4} - 435980 T^{6} + 9671 p^{2} T^{8} - 138 p^{4} T^{10} + p^{6} T^{12} \) |
| 41 | \( 1 - 169 T^{2} + 14057 T^{4} - 714057 T^{6} + 14057 p^{2} T^{8} - 169 p^{4} T^{10} + p^{6} T^{12} \) |
| 43 | \( ( 1 + 11 T + 125 T^{2} + 945 T^{3} + 125 p T^{4} + 11 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 47 | \( 1 - 206 T^{2} + 20175 T^{4} - 1193924 T^{6} + 20175 p^{2} T^{8} - 206 p^{4} T^{10} + p^{6} T^{12} \) |
| 53 | \( ( 1 + 131 T^{2} + 56 T^{3} + 131 p T^{4} + p^{3} T^{6} )^{2} \) |
| 59 | \( 1 - 225 T^{2} + 24609 T^{4} - 1732041 T^{6} + 24609 p^{2} T^{8} - 225 p^{4} T^{10} + p^{6} T^{12} \) |
| 61 | \( ( 1 + 4 T + 39 T^{2} + 424 T^{3} + 39 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 67 | \( 1 - 129 T^{2} + 13281 T^{4} - 1140649 T^{6} + 13281 p^{2} T^{8} - 129 p^{4} T^{10} + p^{6} T^{12} \) |
| 71 | \( 1 - 322 T^{2} + 46287 T^{4} - 4042108 T^{6} + 46287 p^{2} T^{8} - 322 p^{4} T^{10} + p^{6} T^{12} \) |
| 73 | \( 1 - 265 T^{2} + 36313 T^{4} - 3192361 T^{6} + 36313 p^{2} T^{8} - 265 p^{4} T^{10} + p^{6} T^{12} \) |
| 79 | \( ( 1 + 18 T + 261 T^{2} + 2612 T^{3} + 261 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 83 | \( 1 - 465 T^{2} + 92609 T^{4} - 10109729 T^{6} + 92609 p^{2} T^{8} - 465 p^{4} T^{10} + p^{6} T^{12} \) |
| 89 | \( 1 - 97 T^{2} + 9929 T^{4} - 1067577 T^{6} + 9929 p^{2} T^{8} - 97 p^{4} T^{10} + p^{6} T^{12} \) |
| 97 | \( 1 - 177 T^{2} + 28457 T^{4} - 3235529 T^{6} + 28457 p^{2} T^{8} - 177 p^{4} T^{10} + p^{6} T^{12} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.53716738054647788713314899516, −4.47521823440233855903545188470, −4.29389173115548140476434215572, −4.24221186335182560746845484676, −3.83055234760991215357577854435, −3.76811823269179366126800043675, −3.74018401203051118316850230193, −3.30577722963224612401676361697, −3.25116414327503666337515873788, −3.07393260532965826517525980992, −3.06802906373950058718836532445, −3.05021499734504841889550861221, −2.86602282529951307294569035982, −2.56514887054693975881165220339, −2.35101548620972613679528606088, −2.28524599997780949380412394376, −1.91875289099882913266206295708, −1.54821710628673489351598543893, −1.52036336273837836761353273915, −1.41740438803720495397971582615, −1.15070178773017464160242653461, −0.974508859045010877982372206664, −0.62990410583433074933006344720, −0.57759996020165181220393368902, −0.32232188984465835106850935419,
0.32232188984465835106850935419, 0.57759996020165181220393368902, 0.62990410583433074933006344720, 0.974508859045010877982372206664, 1.15070178773017464160242653461, 1.41740438803720495397971582615, 1.52036336273837836761353273915, 1.54821710628673489351598543893, 1.91875289099882913266206295708, 2.28524599997780949380412394376, 2.35101548620972613679528606088, 2.56514887054693975881165220339, 2.86602282529951307294569035982, 3.05021499734504841889550861221, 3.06802906373950058718836532445, 3.07393260532965826517525980992, 3.25116414327503666337515873788, 3.30577722963224612401676361697, 3.74018401203051118316850230193, 3.76811823269179366126800043675, 3.83055234760991215357577854435, 4.24221186335182560746845484676, 4.29389173115548140476434215572, 4.47521823440233855903545188470, 4.53716738054647788713314899516
Plot not available for L-functions of degree greater than 10.
