| L(s) = 1 | − 16·13-s + 16·37-s + 24·49-s − 48·53-s + 16·61-s − 48·101-s + 64·109-s − 48·113-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 128·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + ⋯ |
| L(s) = 1 | − 4.43·13-s + 2.63·37-s + 24/7·49-s − 6.59·53-s + 2.04·61-s − 4.77·101-s + 6.13·109-s − 4.51·113-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 + 9.84·169-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 + 0.0669·223-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{64} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{64} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.798439830\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.798439830\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( ( 1 - 34 T^{4} + p^{4} T^{8} )^{2} \) |
| 7 | \( ( 1 - 12 T^{2} + 86 T^{4} - 12 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 11 | \( 1 - 68 T^{4} - 12762 T^{8} - 68 p^{4} T^{12} + p^{8} T^{16} \) |
| 13 | \( ( 1 + 8 T + 32 T^{2} + 120 T^{3} + 446 T^{4} + 120 p T^{5} + 32 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 17 | \( ( 1 + 22 T^{2} + p^{2} T^{4} )^{4} \) |
| 19 | \( 1 + 124 T^{4} - 27546 T^{8} + 124 p^{4} T^{12} + p^{8} T^{16} \) |
| 23 | \( ( 1 - 44 T^{2} + 1110 T^{4} - 44 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 + 1022 T^{4} + p^{4} T^{8} )^{2} \) |
| 31 | \( ( 1 + 30 T^{2} + p^{2} T^{4} )^{4} \) |
| 37 | \( ( 1 - 8 T + 32 T^{2} - 312 T^{3} + 3038 T^{4} - 312 p T^{5} + 32 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 41 | \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{4} \) |
| 43 | \( 1 - 452 T^{4} - 2831322 T^{8} - 452 p^{4} T^{12} + p^{8} T^{16} \) |
| 47 | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{4} \) |
| 53 | \( ( 1 + 24 T + 288 T^{2} + 2856 T^{3} + 23966 T^{4} + 2856 p T^{5} + 288 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 59 | \( 1 - 9668 T^{4} + 43107750 T^{8} - 9668 p^{4} T^{12} + p^{8} T^{16} \) |
| 61 | \( ( 1 - 8 T + 32 T^{2} - 504 T^{3} + 7934 T^{4} - 504 p T^{5} + 32 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 67 | \( 1 + 4732 T^{4} + 41046246 T^{8} + 4732 p^{4} T^{12} + p^{8} T^{16} \) |
| 71 | \( ( 1 - 236 T^{2} + 23574 T^{4} - 236 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 73 | \( ( 1 - 196 T^{2} + 18534 T^{4} - 196 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 79 | \( ( 1 + 60 T^{2} + 1094 T^{4} + 60 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 83 | \( 1 + 8188 T^{4} + 45970278 T^{8} + 8188 p^{4} T^{12} + p^{8} T^{16} \) |
| 89 | \( ( 1 - 260 T^{2} + 31014 T^{4} - 260 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 97 | \( ( 1 + 182 T^{2} + p^{2} T^{4} )^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.92791037522398125906196650337, −3.62113777319441891378542538415, −3.37728091912251403163529142075, −3.34576915910385019348630522339, −3.14927464931236669965047322727, −3.11529152415550595831881730702, −3.00112614794161338312384515535, −2.92625435318348320299439099048, −2.84854582067871515844140343484, −2.58707262262948767961445394205, −2.53576161463989811807079429743, −2.28750219412718513691306503792, −2.26411567213665219838437667819, −2.14213633678754953676005849547, −2.12413680966248023924674570640, −1.92800029381610741219385327005, −1.72098155555150119195472034498, −1.60603374616514230706322851998, −1.26640962308645660785452489062, −1.09336151599985290382181379845, −0.987190011716098147139464432060, −0.965287679056874727432973979105, −0.40801682636467236999627077774, −0.26675520894466077344090085972, −0.25516467074787652411515879981,
0.25516467074787652411515879981, 0.26675520894466077344090085972, 0.40801682636467236999627077774, 0.965287679056874727432973979105, 0.987190011716098147139464432060, 1.09336151599985290382181379845, 1.26640962308645660785452489062, 1.60603374616514230706322851998, 1.72098155555150119195472034498, 1.92800029381610741219385327005, 2.12413680966248023924674570640, 2.14213633678754953676005849547, 2.26411567213665219838437667819, 2.28750219412718513691306503792, 2.53576161463989811807079429743, 2.58707262262948767961445394205, 2.84854582067871515844140343484, 2.92625435318348320299439099048, 3.00112614794161338312384515535, 3.11529152415550595831881730702, 3.14927464931236669965047322727, 3.34576915910385019348630522339, 3.37728091912251403163529142075, 3.62113777319441891378542538415, 3.92791037522398125906196650337
Plot not available for L-functions of degree greater than 10.
