| L(s) = 1 | + 16·9-s − 32·25-s − 20·49-s + 8·53-s + 128·81-s − 64·97-s − 32·113-s + 6·121-s − 8·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 76·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯ |
| L(s) = 1 | + 16/3·9-s − 6.39·25-s − 2.85·49-s + 1.09·53-s + 14.2·81-s − 6.49·97-s − 3.01·113-s + 6/11·121-s − 0.715·125-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 + 5.84·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{60} \cdot 11^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{60} \cdot 11^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.631504725\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.631504725\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 - 6 T^{2} - 73 T^{4} + 460 T^{6} - 73 p^{2} T^{8} - 6 p^{4} T^{10} + p^{6} T^{12} \) |
| good | 3 | \( ( 1 - 8 T^{2} + 32 T^{4} - 98 T^{6} + 32 p^{2} T^{8} - 8 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 5 | \( ( 1 + 8 T^{2} + 2 T^{3} + 8 p T^{4} + p^{3} T^{6} )^{4} \) |
| 7 | \( ( 1 + 10 T^{2} + 111 T^{4} + 748 T^{6} + 111 p^{2} T^{8} + 10 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 13 | \( ( 1 - 38 T^{2} + 599 T^{4} - 6996 T^{6} + 599 p^{2} T^{8} - 38 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 17 | \( ( 1 - 38 T^{2} + 1279 T^{4} - 23156 T^{6} + 1279 p^{2} T^{8} - 38 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 19 | \( ( 1 + 10 T^{2} + 471 T^{4} + 8908 T^{6} + 471 p^{2} T^{8} + 10 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 23 | \( ( 1 - 88 T^{2} + 4112 T^{4} - 116538 T^{6} + 4112 p^{2} T^{8} - 88 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 29 | \( ( 1 - 38 T^{2} + 935 T^{4} - 6324 T^{6} + 935 p^{2} T^{8} - 38 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 31 | \( ( 1 - 88 T^{2} + 3632 T^{4} - 114474 T^{6} + 3632 p^{2} T^{8} - 88 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 37 | \( ( 1 + 104 T^{2} + 2 T^{3} + 104 p T^{4} + p^{3} T^{6} )^{4} \) |
| 41 | \( ( 1 - 86 T^{2} + 7439 T^{4} - 310740 T^{6} + 7439 p^{2} T^{8} - 86 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 43 | \( ( 1 + 186 T^{2} + 16631 T^{4} + 899404 T^{6} + 16631 p^{2} T^{8} + 186 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 47 | \( ( 1 - 126 T^{2} + 11471 T^{4} - 610916 T^{6} + 11471 p^{2} T^{8} - 126 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 53 | \( ( 1 - 2 T + 91 T^{2} - 204 T^{3} + 91 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} )^{4} \) |
| 59 | \( ( 1 - 168 T^{2} + 14304 T^{4} - 927858 T^{6} + 14304 p^{2} T^{8} - 168 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 61 | \( ( 1 - 230 T^{2} + 24935 T^{4} - 1782324 T^{6} + 24935 p^{2} T^{8} - 230 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 67 | \( ( 1 - 200 T^{2} + 24448 T^{4} - 1950914 T^{6} + 24448 p^{2} T^{8} - 200 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 71 | \( ( 1 - 184 T^{2} + 14576 T^{4} - 905562 T^{6} + 14576 p^{2} T^{8} - 184 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 73 | \( ( 1 - 278 T^{2} + 41679 T^{4} - 3752276 T^{6} + 41679 p^{2} T^{8} - 278 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 79 | \( ( 1 + 30 T^{2} + p^{2} T^{4} )^{6} \) |
| 83 | \( ( 1 + 234 T^{2} + 30023 T^{4} + 2772268 T^{6} + 30023 p^{2} T^{8} + 234 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 89 | \( ( 1 + 80 T^{2} + 358 T^{3} + 80 p T^{4} + p^{3} T^{6} )^{4} \) |
| 97 | \( ( 1 + 16 T + 272 T^{2} + 2382 T^{3} + 272 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} )^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{24} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.97476265155392712654219042608, −3.79507747067072158836167691309, −3.74233511904721328525009843824, −3.70726584608862198874254727620, −3.51105197031136079346366858477, −3.41602910432500097165539468459, −3.32012128631068148055953057266, −3.10495772674792059380317482392, −3.07471915989349300059710040789, −2.94783225097990509282968483657, −2.65849948905454806325168549009, −2.56180405490412421782998039286, −2.36684652880227041738922267683, −2.21352439853079373817864203079, −2.12569216467587796842777519458, −2.09926067505529286289616881849, −1.82470409520826490759343552209, −1.80826843135226397174217596901, −1.53575223585370216916168653561, −1.47946574280253193631971354326, −1.37195063560448671364472816898, −1.26802448062448540270173985493, −1.16751638528129477921871070377, −0.49560360651987670943242247853, −0.30868528919688185645144879745,
0.30868528919688185645144879745, 0.49560360651987670943242247853, 1.16751638528129477921871070377, 1.26802448062448540270173985493, 1.37195063560448671364472816898, 1.47946574280253193631971354326, 1.53575223585370216916168653561, 1.80826843135226397174217596901, 1.82470409520826490759343552209, 2.09926067505529286289616881849, 2.12569216467587796842777519458, 2.21352439853079373817864203079, 2.36684652880227041738922267683, 2.56180405490412421782998039286, 2.65849948905454806325168549009, 2.94783225097990509282968483657, 3.07471915989349300059710040789, 3.10495772674792059380317482392, 3.32012128631068148055953057266, 3.41602910432500097165539468459, 3.51105197031136079346366858477, 3.70726584608862198874254727620, 3.74233511904721328525009843824, 3.79507747067072158836167691309, 3.97476265155392712654219042608
Plot not available for L-functions of degree greater than 10.
