| L(s) = 1 | + 4-s + 9-s + 12·13-s + 3·16-s + 36·25-s + 36-s + 8·37-s + 12·52-s + 4·61-s − 3·64-s − 6·81-s − 36·97-s + 36·100-s − 32·109-s + 12·117-s − 88·121-s + 127-s + 131-s + 137-s + 139-s + 3·144-s + 8·148-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | + 1/2·4-s + 1/3·9-s + 3.32·13-s + 3/4·16-s + 36/5·25-s + 1/6·36-s + 1.31·37-s + 1.66·52-s + 0.512·61-s − 3/8·64-s − 2/3·81-s − 3.65·97-s + 18/5·100-s − 3.06·109-s + 1.10·117-s − 8·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 1/4·144-s + 0.657·148-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{12} \cdot 7^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{12} \cdot 7^{24}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(16.16088098\) |
| \(L(\frac12)\) |
\(\approx\) |
\(16.16088098\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T^{2} - p T^{4} + p^{3} T^{6} - p^{3} T^{8} - p^{4} T^{10} + p^{6} T^{12} \) |
| 3 | \( 1 - T^{2} + 7 T^{4} - 14 T^{6} + 7 p^{2} T^{8} - p^{4} T^{10} + p^{6} T^{12} \) |
| 7 | \( 1 \) |
| good | 5 | \( ( 1 - 18 T^{2} + 34 p T^{4} - 1042 T^{6} + 34 p^{3} T^{8} - 18 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 11 | \( ( 1 + 4 p T^{2} + 952 T^{4} + 12830 T^{6} + 952 p^{2} T^{8} + 4 p^{5} T^{10} + p^{6} T^{12} )^{2} \) |
| 13 | \( ( 1 - 3 T + 29 T^{2} - 62 T^{3} + 29 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} )^{4} \) |
| 17 | \( ( 1 - 59 T^{2} + 1723 T^{4} - 34178 T^{6} + 1723 p^{2} T^{8} - 59 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 19 | \( ( 1 - 98 T^{2} + 4270 T^{4} - 105158 T^{6} + 4270 p^{2} T^{8} - 98 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 23 | \( ( 1 + 55 T^{2} + 1831 T^{4} + 48802 T^{6} + 1831 p^{2} T^{8} + 55 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 29 | \( ( 1 - 129 T^{2} + 7599 T^{4} - 272414 T^{6} + 7599 p^{2} T^{8} - 129 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 31 | \( ( 1 - 115 T^{2} + 6850 T^{4} - 257011 T^{6} + 6850 p^{2} T^{8} - 115 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 37 | \( ( 1 - 2 T + 32 T^{2} + 144 T^{3} + 32 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} )^{4} \) |
| 41 | \( ( 1 - 102 T^{2} + 8303 T^{4} - 375412 T^{6} + 8303 p^{2} T^{8} - 102 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 43 | \( ( 1 - 227 T^{2} + 22639 T^{4} - 1266458 T^{6} + 22639 p^{2} T^{8} - 227 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 47 | \( ( 1 + 199 T^{2} + 19063 T^{4} + 1118818 T^{6} + 19063 p^{2} T^{8} + 199 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 53 | \( ( 1 - 154 T^{2} + 13858 T^{4} - 826258 T^{6} + 13858 p^{2} T^{8} - 154 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 59 | \( ( 1 + 308 T^{2} + 712 p T^{4} + 3220518 T^{6} + 712 p^{3} T^{8} + 308 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 61 | \( ( 1 - T + 163 T^{2} - 118 T^{3} + 163 p T^{4} - p^{2} T^{5} + p^{3} T^{6} )^{4} \) |
| 67 | \( ( 1 - 258 T^{2} + 32174 T^{4} - 2581142 T^{6} + 32174 p^{2} T^{8} - 258 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 71 | \( ( 1 + 374 T^{2} + 61519 T^{4} + 5680340 T^{6} + 61519 p^{2} T^{8} + 374 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 73 | \( ( 1 + 80 T^{2} + 394 T^{3} + 80 p T^{4} + p^{3} T^{6} )^{4} \) |
| 79 | \( ( 1 - 243 T^{2} + 34434 T^{4} - 3335443 T^{6} + 34434 p^{2} T^{8} - 243 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 83 | \( ( 1 + 115 T^{2} + 23611 T^{4} + 1604482 T^{6} + 23611 p^{2} T^{8} + 115 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 89 | \( ( 1 - 455 T^{2} + 91151 T^{4} - 10426578 T^{6} + 91151 p^{2} T^{8} - 455 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 97 | \( ( 1 + 9 T + 189 T^{2} + 1690 T^{3} + 189 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} )^{4} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{24} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.49637228791575362722761031792, −3.35692249578913521259294234317, −3.23443628606715428426720594420, −3.11395638135808044408553156109, −3.06070883362744523026642109375, −2.95174158830476769767275682298, −2.85043622773485852751029107298, −2.84200850840767786529063229693, −2.76727737874650459432038790991, −2.66536741124366375391386997480, −2.54878069647133526291931261927, −2.36689995271946033325933825208, −2.35689056761999694313574828080, −2.00086475876803871196818615934, −1.94544828574442291092119880560, −1.68718188379239668647319059128, −1.49699081777781396187069506090, −1.44825054827152608074476316123, −1.36495102300101663330535592059, −1.28383194950219184960404184037, −1.13306658624469536015575344761, −1.00804588519643503859310579725, −0.802591594391509387108688329782, −0.68881037054859609991264609461, −0.29628708589438167231760017345,
0.29628708589438167231760017345, 0.68881037054859609991264609461, 0.802591594391509387108688329782, 1.00804588519643503859310579725, 1.13306658624469536015575344761, 1.28383194950219184960404184037, 1.36495102300101663330535592059, 1.44825054827152608074476316123, 1.49699081777781396187069506090, 1.68718188379239668647319059128, 1.94544828574442291092119880560, 2.00086475876803871196818615934, 2.35689056761999694313574828080, 2.36689995271946033325933825208, 2.54878069647133526291931261927, 2.66536741124366375391386997480, 2.76727737874650459432038790991, 2.84200850840767786529063229693, 2.85043622773485852751029107298, 2.95174158830476769767275682298, 3.06070883362744523026642109375, 3.11395638135808044408553156109, 3.23443628606715428426720594420, 3.35692249578913521259294234317, 3.49637228791575362722761031792
Plot not available for L-functions of degree greater than 10.
