| L(s) = 1 | + 16·7-s − 8·13-s + 64·19-s + 10·25-s − 8·31-s − 272·37-s − 80·43-s + 112·49-s + 40·61-s + 304·67-s + 304·73-s − 200·79-s − 128·91-s + 424·97-s + 112·103-s + 208·109-s − 88·121-s + 127-s + 131-s + 1.02e3·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | + 16/7·7-s − 0.615·13-s + 3.36·19-s + 2/5·25-s − 0.258·31-s − 7.35·37-s − 1.86·43-s + 16/7·49-s + 0.655·61-s + 4.53·67-s + 4.16·73-s − 2.53·79-s − 1.40·91-s + 4.37·97-s + 1.08·103-s + 1.90·109-s − 0.727·121-s + 0.00787·127-s + 0.00763·131-s + 7.69·133-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{32} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{32} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.654559155\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.654559155\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| good | 7 | \( ( 1 - 8 T + 40 T^{2} + 592 T^{3} - 4961 T^{4} + 592 p^{2} T^{5} + 40 p^{4} T^{6} - 8 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 11 | \( 1 + 8 p T^{2} - 10514 T^{4} - 88192 p T^{6} + 5615779 T^{8} - 88192 p^{5} T^{10} - 10514 p^{8} T^{12} + 8 p^{13} T^{14} + p^{16} T^{16} \) |
| 13 | \( ( 1 + 4 T - 236 T^{2} - 344 T^{3} + 32239 T^{4} - 344 p^{2} T^{5} - 236 p^{4} T^{6} + 4 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 17 | \( ( 1 - 290 T^{2} + p^{4} T^{4} )^{4} \) |
| 19 | \( ( 1 - 16 T + 426 T^{2} - 16 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 23 | \( 1 + 1612 T^{2} + 1441066 T^{4} + 963647152 T^{6} + 534265183699 T^{8} + 963647152 p^{4} T^{10} + 1441066 p^{8} T^{12} + 1612 p^{12} T^{14} + p^{16} T^{16} \) |
| 29 | \( 1 + 700 T^{2} - 217622 T^{4} - 494858000 T^{6} - 346085208077 T^{8} - 494858000 p^{4} T^{10} - 217622 p^{8} T^{12} + 700 p^{12} T^{14} + p^{16} T^{16} \) |
| 31 | \( ( 1 + 4 T - 50 p T^{2} - 1424 T^{3} + 1513459 T^{4} - 1424 p^{2} T^{5} - 50 p^{5} T^{6} + 4 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 37 | \( ( 1 + 68 T + 3084 T^{2} + 68 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 41 | \( 1 + 640 T^{2} - 3478082 T^{4} - 1128857600 T^{6} + 5971714398403 T^{8} - 1128857600 p^{4} T^{10} - 3478082 p^{8} T^{12} + 640 p^{12} T^{14} + p^{16} T^{16} \) |
| 43 | \( ( 1 + 40 T - 2138 T^{2} + 1600 T^{3} + 7595443 T^{4} + 1600 p^{2} T^{5} - 2138 p^{4} T^{6} + 40 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 47 | \( 1 + 340 T^{2} + 487978 T^{4} - 3444791600 T^{6} - 24236712410477 T^{8} - 3444791600 p^{4} T^{10} + 487978 p^{8} T^{12} + 340 p^{12} T^{14} + p^{16} T^{16} \) |
| 53 | \( ( 1 - 1876 T^{2} - 4075194 T^{4} - 1876 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 59 | \( 1 + 280 T^{2} + 9533038 T^{4} - 9433020800 T^{6} - 58390258995677 T^{8} - 9433020800 p^{4} T^{10} + 9533038 p^{8} T^{12} + 280 p^{12} T^{14} + p^{16} T^{16} \) |
| 61 | \( ( 1 - 20 T - 3902 T^{2} + 62800 T^{3} + 3172963 T^{4} + 62800 p^{2} T^{5} - 3902 p^{4} T^{6} - 20 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 67 | \( ( 1 - 76 T + 1287 T^{2} - 76 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 71 | \( ( 1 - 13540 T^{2} + 89191302 T^{4} - 13540 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 73 | \( ( 1 - 76 T + 8862 T^{2} - 76 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 79 | \( ( 1 + 100 T - 1742 T^{2} - 74000 T^{3} + 36514483 T^{4} - 74000 p^{2} T^{5} - 1742 p^{4} T^{6} + 100 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 83 | \( 1 + 21652 T^{2} + 257519626 T^{4} + 2519704645072 T^{6} + 20142058336019059 T^{8} + 2519704645072 p^{4} T^{10} + 257519626 p^{8} T^{12} + 21652 p^{12} T^{14} + p^{16} T^{16} \) |
| 89 | \( ( 1 - 14368 T^{2} + 154232898 T^{4} - 14368 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 97 | \( ( 1 - 212 T + 15250 T^{2} - 2305712 T^{3} + 370326259 T^{4} - 2305712 p^{2} T^{5} + 15250 p^{4} T^{6} - 212 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
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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
−3.63081315697167324119024482935, −3.49818991654411969119937000733, −3.49685598436819181951404365539, −3.48592237570551463113399505813, −3.46820454802401170536619750071, −3.18987509467134113428894112702, −3.03124802122008857099510799658, −2.80161635534760173254375591955, −2.67589824507794425130572866760, −2.66855334073935105145791396721, −2.31633340594718057239299056996, −2.31502754981315103068493336353, −2.05339449871388913259302626344, −1.87890952806688220399252081192, −1.83337943415226365966693255257, −1.81721353566779621917536735162, −1.58514509015207029432679017934, −1.41242162094055244958471098050, −1.40840800049109976359730899775, −1.06893712735736260524007389107, −0.75902945607078682776009434895, −0.75842351514920800694307781105, −0.70381405719877988101267905707, −0.35051889138418589714406698141, −0.079689813120561900714951891034,
0.079689813120561900714951891034, 0.35051889138418589714406698141, 0.70381405719877988101267905707, 0.75842351514920800694307781105, 0.75902945607078682776009434895, 1.06893712735736260524007389107, 1.40840800049109976359730899775, 1.41242162094055244958471098050, 1.58514509015207029432679017934, 1.81721353566779621917536735162, 1.83337943415226365966693255257, 1.87890952806688220399252081192, 2.05339449871388913259302626344, 2.31502754981315103068493336353, 2.31633340594718057239299056996, 2.66855334073935105145791396721, 2.67589824507794425130572866760, 2.80161635534760173254375591955, 3.03124802122008857099510799658, 3.18987509467134113428894112702, 3.46820454802401170536619750071, 3.48592237570551463113399505813, 3.49685598436819181951404365539, 3.49818991654411969119937000733, 3.63081315697167324119024482935
Plot not available for L-functions of degree greater than 10.
