| L(s) = 1 | − 72·9-s − 60·17-s − 62·25-s − 146·49-s − 100·73-s + 2.91e3·81-s − 466·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 4.32e3·153-s + 157-s + 163-s + 167-s − 1.35e3·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + ⋯ |
| L(s) = 1 | − 8·9-s − 3.52·17-s − 2.47·25-s − 2.97·49-s − 1.36·73-s + 36·81-s − 3.85·121-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 + 28.2·153-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s − 8·169-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + 0.00507·197-s + 0.00502·199-s + 0.00473·211-s + 0.00448·223-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.002001655529\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.002001655529\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( ( 1 + p^{2} T^{2} )^{4} \) |
| good | 3 | \( ( 1 + p^{2} T^{2} )^{8} \) |
| 5 | \( ( 1 - 9 T + 56 T^{2} - 9 p^{2} T^{3} + p^{4} T^{4} )^{2}( 1 + 9 T + 56 T^{2} + 9 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 7 | \( ( 1 + 73 T^{2} + 2928 T^{4} + 73 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 11 | \( ( 1 + 233 T^{2} + 39648 T^{4} + 233 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 13 | \( ( 1 + p^{2} T^{2} )^{8} \) |
| 17 | \( ( 1 + 15 T - 64 T^{2} + 15 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 23 | \( ( 1 - 158 T^{2} + p^{4} T^{4} )^{4} \) |
| 29 | \( ( 1 + p^{2} T^{2} )^{8} \) |
| 31 | \( ( 1 - p T )^{8}( 1 + p T )^{8} \) |
| 37 | \( ( 1 + p^{2} T^{2} )^{8} \) |
| 41 | \( ( 1 - p T )^{8}( 1 + p T )^{8} \) |
| 43 | \( ( 1 - 3527 T^{2} + 9020928 T^{4} - 3527 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 47 | \( ( 1 - 1207 T^{2} - 3422832 T^{4} - 1207 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 53 | \( ( 1 + p^{2} T^{2} )^{8} \) |
| 59 | \( ( 1 + p^{2} T^{2} )^{8} \) |
| 61 | \( ( 1 - 103 T + 6888 T^{2} - 103 p^{2} T^{3} + p^{4} T^{4} )^{2}( 1 + 103 T + 6888 T^{2} + 103 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 67 | \( ( 1 + p^{2} T^{2} )^{8} \) |
| 71 | \( ( 1 - p T )^{8}( 1 + p T )^{8} \) |
| 73 | \( ( 1 + 25 T - 4704 T^{2} + 25 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 79 | \( ( 1 - p T )^{8}( 1 + p T )^{8} \) |
| 83 | \( ( 1 - 5678 T^{2} + p^{4} T^{4} )^{4} \) |
| 89 | \( ( 1 - p T )^{8}( 1 + p T )^{8} \) |
| 97 | \( ( 1 - p T )^{8}( 1 + p T )^{8} \) |
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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
−3.80580953289810200051144127071, −3.78987573426375244784324605424, −3.68482589359772244903796971184, −3.52598614014140722800648326154, −3.45377199291649519676308751576, −3.16313575748285918876117213038, −2.95886932010051714172849050168, −2.86609283514675017240011422744, −2.83405857441084593332164502992, −2.79065224034652483393403183579, −2.66077533580069245585920641383, −2.44438121016698162500827524811, −2.36653901302181592267897811152, −2.29021377464043412122419509728, −2.12563377435366652091667137808, −1.92029022163731088158712209538, −1.80154585676754745759645163154, −1.65860974424051416301467395533, −1.23395014084039058477960692535, −1.10254222728335197246984775478, −0.845122832151694168805711418218, −0.39943775068056462498006173879, −0.28302573297101250628464372836, −0.11735679930329908895961527470, −0.03412254195990647798593440929,
0.03412254195990647798593440929, 0.11735679930329908895961527470, 0.28302573297101250628464372836, 0.39943775068056462498006173879, 0.845122832151694168805711418218, 1.10254222728335197246984775478, 1.23395014084039058477960692535, 1.65860974424051416301467395533, 1.80154585676754745759645163154, 1.92029022163731088158712209538, 2.12563377435366652091667137808, 2.29021377464043412122419509728, 2.36653901302181592267897811152, 2.44438121016698162500827524811, 2.66077533580069245585920641383, 2.79065224034652483393403183579, 2.83405857441084593332164502992, 2.86609283514675017240011422744, 2.95886932010051714172849050168, 3.16313575748285918876117213038, 3.45377199291649519676308751576, 3.52598614014140722800648326154, 3.68482589359772244903796971184, 3.78987573426375244784324605424, 3.80580953289810200051144127071
Plot not available for L-functions of degree greater than 10.
