| L(s) = 1 | + 16·5-s − 48·13-s − 96·17-s + 128·25-s + 80·37-s + 400·41-s + 96·53-s − 144·61-s − 768·65-s + 120·73-s − 18·81-s − 1.53e3·85-s − 360·97-s + 320·101-s + 176·113-s − 368·121-s + 912·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 1.15e3·169-s + ⋯ |
| L(s) = 1 | + 16/5·5-s − 3.69·13-s − 5.64·17-s + 5.11·25-s + 2.16·37-s + 9.75·41-s + 1.81·53-s − 2.36·61-s − 11.8·65-s + 1.64·73-s − 2/9·81-s − 18.0·85-s − 3.71·97-s + 3.16·101-s + 1.55·113-s − 3.04·121-s + 7.29·125-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 6.81·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(11.09553211\) |
| \(L(\frac12)\) |
\(\approx\) |
\(11.09553211\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 5 | \( ( 1 - 8 T + 32 T^{2} - 8 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| good | 7 | \( ( 1 - 4786 T^{4} + p^{8} T^{8} )^{2} \) |
| 11 | \( ( 1 + 92 T^{2} + p^{4} T^{4} )^{4} \) |
| 13 | \( ( 1 + 24 T + 288 T^{2} + 4152 T^{3} + 59842 T^{4} + 4152 p^{2} T^{5} + 288 p^{4} T^{6} + 24 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 17 | \( ( 1 + 48 T + 1152 T^{2} + 24432 T^{3} + 469762 T^{4} + 24432 p^{2} T^{5} + 1152 p^{4} T^{6} + 48 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 19 | \( ( 1 - 52 p T^{2} + 485094 T^{4} - 52 p^{5} T^{6} + p^{8} T^{8} )^{2} \) |
| 23 | \( ( 1 + 342818 T^{4} + p^{8} T^{8} )^{2} \) |
| 29 | \( ( 1 + 96 T^{2} + 990370 T^{4} + 96 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 31 | \( ( 1 + 1538 T^{2} + p^{4} T^{4} )^{4} \) |
| 37 | \( ( 1 - 40 T + 800 T^{2} - 38280 T^{3} + 1661954 T^{4} - 38280 p^{2} T^{5} + 800 p^{4} T^{6} - 40 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 41 | \( ( 1 - 100 T + 5726 T^{2} - 100 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 43 | \( 1 - 12264188 T^{4} + 60977612453574 T^{8} - 12264188 p^{8} T^{12} + p^{16} T^{16} \) |
| 47 | \( 1 + 2112772 T^{4} - 2013306 p^{4} T^{8} + 2112772 p^{8} T^{12} + p^{16} T^{16} \) |
| 53 | \( ( 1 - 48 T + 1152 T^{2} - 96432 T^{3} + 7432162 T^{4} - 96432 p^{2} T^{5} + 1152 p^{4} T^{6} - 48 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 59 | \( ( 1 + 1352 T^{2} + 16055154 T^{4} + 1352 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 61 | \( ( 1 + 36 T + 6542 T^{2} + 36 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 67 | \( 1 - 10015484 T^{4} + 340863405291846 T^{8} - 10015484 p^{8} T^{12} + p^{16} T^{16} \) |
| 71 | \( ( 1 + 10564 T^{2} + 58668870 T^{4} + 10564 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 73 | \( ( 1 - 60 T + 1800 T^{2} + 61260 T^{3} - 38237618 T^{4} + 61260 p^{2} T^{5} + 1800 p^{4} T^{6} - 60 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 79 | \( ( 1 + 4316 T^{2} + 37435590 T^{4} + 4316 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 83 | \( ( 1 - 5689726 T^{4} + p^{8} T^{8} )^{2} \) |
| 89 | \( ( 1 - 25980 T^{2} + 288340678 T^{4} - 25980 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 97 | \( ( 1 + 180 T + 16200 T^{2} + 2226780 T^{3} + 297309838 T^{4} + 2226780 p^{2} T^{5} + 16200 p^{4} T^{6} + 180 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
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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
−4.56368741056611221616615388944, −4.40794404210556156476493835170, −4.35584442321485839222839749102, −4.27574359036723554177427702609, −4.26204416646679084275550118377, −3.99607346445429354268986124622, −3.85087832432222126403561658199, −3.74772561804689069446521365286, −3.23851395650119794535516687703, −2.89105702519059276285919221023, −2.85688941152264007647808545895, −2.68986478235766760679933900823, −2.63827346986282786719446530482, −2.61023935881811811526880513283, −2.20191448398529664814698115349, −2.19806269542178577767176064584, −2.19019124387184278963803020212, −2.12113689548570494461951382397, −2.06414975510780718747035596053, −1.39794802933172010512467364306, −1.25586440800127659760103580538, −1.09956300167855730472719404565, −0.52354959734995923514505136466, −0.48331125007526916813840674543, −0.37565148932902724607763152263,
0.37565148932902724607763152263, 0.48331125007526916813840674543, 0.52354959734995923514505136466, 1.09956300167855730472719404565, 1.25586440800127659760103580538, 1.39794802933172010512467364306, 2.06414975510780718747035596053, 2.12113689548570494461951382397, 2.19019124387184278963803020212, 2.19806269542178577767176064584, 2.20191448398529664814698115349, 2.61023935881811811526880513283, 2.63827346986282786719446530482, 2.68986478235766760679933900823, 2.85688941152264007647808545895, 2.89105702519059276285919221023, 3.23851395650119794535516687703, 3.74772561804689069446521365286, 3.85087832432222126403561658199, 3.99607346445429354268986124622, 4.26204416646679084275550118377, 4.27574359036723554177427702609, 4.35584442321485839222839749102, 4.40794404210556156476493835170, 4.56368741056611221616615388944
Plot not available for L-functions of degree greater than 10.
