| L(s) = 1 | − 2·3-s − 3·4-s − 11·9-s + 6·12-s + 6·16-s + 2·17-s + 20·23-s − 3·25-s + 28·27-s − 16·29-s + 33·36-s − 10·43-s − 12·48-s + 22·49-s − 4·51-s − 16·53-s − 16·61-s − 10·64-s − 6·68-s − 40·69-s + 6·75-s + 4·79-s + 56·81-s + 32·87-s − 60·92-s + 9·100-s − 28·101-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 3/2·4-s − 3.66·9-s + 1.73·12-s + 3/2·16-s + 0.485·17-s + 4.17·23-s − 3/5·25-s + 5.38·27-s − 2.97·29-s + 11/2·36-s − 1.52·43-s − 1.73·48-s + 22/7·49-s − 0.560·51-s − 2.19·53-s − 2.04·61-s − 5/4·64-s − 0.727·68-s − 4.81·69-s + 0.692·75-s + 0.450·79-s + 56/9·81-s + 3.43·87-s − 6.25·92-s + 9/10·100-s − 2.78·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 5^{6} \cdot 13^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 5^{6} \cdot 13^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3725029640\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3725029640\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 + T^{2} )^{3} \) |
| 5 | \( ( 1 + T^{2} )^{3} \) |
| 13 | \( 1 \) |
| good | 3 | \( ( 1 + T + 7 T^{2} + 5 T^{3} + 7 p T^{4} + p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 7 | \( 1 - 22 T^{2} + 271 T^{4} - 2260 T^{6} + 271 p^{2} T^{8} - 22 p^{4} T^{10} + p^{6} T^{12} \) |
| 11 | \( 1 - 5 T^{2} + 201 T^{4} - 833 T^{6} + 201 p^{2} T^{8} - 5 p^{4} T^{10} + p^{6} T^{12} \) |
| 17 | \( ( 1 - T + 35 T^{2} - 5 T^{3} + 35 p T^{4} - p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 19 | \( 1 - 97 T^{2} + 4133 T^{4} - 100737 T^{6} + 4133 p^{2} T^{8} - 97 p^{4} T^{10} + p^{6} T^{12} \) |
| 23 | \( ( 1 - 10 T + 93 T^{2} - 468 T^{3} + 93 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 29 | \( ( 1 + 8 T + 99 T^{2} + 456 T^{3} + 99 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 31 | \( 1 - 50 T^{2} + 97 p T^{4} - 96092 T^{6} + 97 p^{3} T^{8} - 50 p^{4} T^{10} + p^{6} T^{12} \) |
| 37 | \( 1 - 42 T^{2} + 1671 T^{4} - 63308 T^{6} + 1671 p^{2} T^{8} - 42 p^{4} T^{10} + p^{6} T^{12} \) |
| 41 | \( 1 - 65 T^{2} - 663 T^{4} + 147823 T^{6} - 663 p^{2} T^{8} - 65 p^{4} T^{10} + p^{6} T^{12} \) |
| 43 | \( ( 1 + 5 T - 5 T^{2} - 409 T^{3} - 5 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 47 | \( 1 - 130 T^{2} + 6623 T^{4} - 255804 T^{6} + 6623 p^{2} T^{8} - 130 p^{4} T^{10} + p^{6} T^{12} \) |
| 53 | \( ( 1 + 8 T + 59 T^{2} + 280 T^{3} + 59 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 59 | \( 1 - 61 T^{2} + 9413 T^{4} - 392889 T^{6} + 9413 p^{2} T^{8} - 61 p^{4} T^{10} + p^{6} T^{12} \) |
| 61 | \( ( 1 + 8 T + 83 T^{2} + 408 T^{3} + 83 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 67 | \( 1 - 117 T^{2} + 10001 T^{4} - 578753 T^{6} + 10001 p^{2} T^{8} - 117 p^{4} T^{10} + p^{6} T^{12} \) |
| 71 | \( 1 - 210 T^{2} + 23439 T^{4} - 1819356 T^{6} + 23439 p^{2} T^{8} - 210 p^{4} T^{10} + p^{6} T^{12} \) |
| 73 | \( 1 - 201 T^{2} + 389 p T^{4} - 2368289 T^{6} + 389 p^{3} T^{8} - 201 p^{4} T^{10} + p^{6} T^{12} \) |
| 79 | \( ( 1 - 2 T + 173 T^{2} - 420 T^{3} + 173 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 83 | \( 1 - 253 T^{2} + 28757 T^{4} - 2390193 T^{6} + 28757 p^{2} T^{8} - 253 p^{4} T^{10} + p^{6} T^{12} \) |
| 89 | \( 1 - 289 T^{2} + 43257 T^{4} - 4528945 T^{6} + 43257 p^{2} T^{8} - 289 p^{4} T^{10} + p^{6} T^{12} \) |
| 97 | \( 1 - 421 T^{2} + 86213 T^{4} - 10545921 T^{6} + 86213 p^{2} T^{8} - 421 p^{4} T^{10} + p^{6} T^{12} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.92040110099039598013352915368, −4.86701315456114480160319986878, −4.63939336927013138508746869431, −4.57162155230570251890003814003, −4.45381374644354356313776306548, −4.05455403171601979152171368282, −3.98691941864725423280219113893, −3.78485351321137717738388372971, −3.48213663614615945758403363601, −3.41307013190645380342890608573, −3.29141717899089638848997219859, −3.10664361649752164237195956476, −3.07782563633889286262413860638, −2.82689892071966083104489236450, −2.75554475256372367224902261102, −2.43423266162024440930611983495, −2.34208882332719444756270001993, −1.89247721694668571090948578894, −1.84370188465203423078209376130, −1.45906640001386034506929974459, −1.28871825476487874960207139503, −0.820783874216809063742439820824, −0.74996660280396308587393737892, −0.32900509271534867055579631166, −0.23431519910518906435011759527,
0.23431519910518906435011759527, 0.32900509271534867055579631166, 0.74996660280396308587393737892, 0.820783874216809063742439820824, 1.28871825476487874960207139503, 1.45906640001386034506929974459, 1.84370188465203423078209376130, 1.89247721694668571090948578894, 2.34208882332719444756270001993, 2.43423266162024440930611983495, 2.75554475256372367224902261102, 2.82689892071966083104489236450, 3.07782563633889286262413860638, 3.10664361649752164237195956476, 3.29141717899089638848997219859, 3.41307013190645380342890608573, 3.48213663614615945758403363601, 3.78485351321137717738388372971, 3.98691941864725423280219113893, 4.05455403171601979152171368282, 4.45381374644354356313776306548, 4.57162155230570251890003814003, 4.63939336927013138508746869431, 4.86701315456114480160319986878, 4.92040110099039598013352915368
Plot not available for L-functions of degree greater than 10.
