| L(s) = 1 | + 800·25-s + 80·31-s − 560·37-s + 2.12e3·49-s − 2.08e3·67-s + 8.80e3·97-s + 676·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 1.40e3·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + ⋯ |
| L(s) = 1 | + 32/5·25-s + 0.463·31-s − 2.48·37-s + 6.18·49-s − 3.79·67-s + 9.21·97-s + 0.507·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s + 0.637·169-s + 0.000439·173-s + 0.000417·179-s + 0.000410·181-s + 0.000378·191-s + 0.000372·193-s + 0.000361·197-s + 0.000356·199-s + 0.000326·211-s + 0.000300·223-s + 0.000292·227-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{16} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{16} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(66.66467740\) |
| \(L(\frac12)\) |
\(\approx\) |
\(66.66467740\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 11 | \( 1 - 676 T^{2} - 14154 p^{2} T^{4} - 676 p^{6} T^{6} + p^{12} T^{8} \) |
| good | 5 | \( ( 1 - 8 p^{2} T^{2} + p^{6} T^{4} )^{4} \) |
| 7 | \( ( 1 - 1060 T^{2} + 514050 T^{4} - 1060 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 13 | \( ( 1 - 700 T^{2} + 5230950 T^{4} - 700 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 17 | \( ( 1 + 6080 T^{2} + 14812350 T^{4} + 6080 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 19 | \( ( 1 - 18436 T^{2} + 166980786 T^{4} - 18436 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 23 | \( ( 1 + 9932 T^{2} + 57602934 T^{4} + 9932 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 29 | \( ( 1 + 69056 T^{2} + 2380486926 T^{4} + 69056 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 31 | \( ( 1 - 20 T + 40350 T^{2} - 20 p^{3} T^{3} + p^{6} T^{4} )^{4} \) |
| 37 | \( ( 1 + 140 T + 52506 T^{2} + 140 p^{3} T^{3} + p^{6} T^{4} )^{4} \) |
| 41 | \( ( 1 + 163184 T^{2} + 15447280446 T^{4} + 163184 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 43 | \( ( 1 - 240340 T^{2} + 26312850450 T^{4} - 240340 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 47 | \( ( 1 - 149846 T^{2} + p^{6} T^{4} )^{4} \) |
| 53 | \( ( 1 - 380608 T^{2} + 80522854674 T^{4} - 380608 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 59 | \( ( 1 - 314500 T^{2} + 97626686982 T^{4} - 314500 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 61 | \( ( 1 - 377524 T^{2} + 127308921366 T^{4} - 377524 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 67 | \( ( 1 + 260 T + p^{3} T^{2} )^{8} \) |
| 71 | \( ( 1 - 978700 T^{2} + 444510659142 T^{4} - 978700 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 73 | \( ( 1 + 379940 T^{2} + 218125296150 T^{4} + 379940 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 79 | \( ( 1 - 160756 T^{2} + 483387051426 T^{4} - 160756 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 83 | \( ( 1 + 1970780 T^{2} + 1609400316150 T^{4} + 1970780 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 89 | \( ( 1 - 1401424 T^{2} + 982134568866 T^{4} - 1401424 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 97 | \( ( 1 - 2200 T + 2552046 T^{2} - 2200 p^{3} T^{3} + p^{6} T^{4} )^{4} \) |
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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
−3.59928833084004307463695668842, −3.49848608694700097849450065210, −3.16614058657274622918266973248, −3.13976128509321437329012400558, −2.96643102865031711709677636882, −2.95907924462993947802819148264, −2.90932818050755766942528050050, −2.86622465931618096063060486825, −2.65061006043583440151297038900, −2.27729605181723478609908773447, −2.19456323228185430533644901576, −2.13568900946397547578514621308, −1.98285334386829952390424308470, −1.90746687134987415547704344010, −1.89566026897872690270131315116, −1.43982470688467799063703792606, −1.35127696561955395342231340934, −1.22841609319962990833565361447, −0.918506323632207175822646229596, −0.868205271215803731438003474297, −0.77879254464508489980220055189, −0.63554898169851134851581969337, −0.60447644684343254246441405314, −0.33821957057162126981939201281, −0.33036303996992563970331016409,
0.33036303996992563970331016409, 0.33821957057162126981939201281, 0.60447644684343254246441405314, 0.63554898169851134851581969337, 0.77879254464508489980220055189, 0.868205271215803731438003474297, 0.918506323632207175822646229596, 1.22841609319962990833565361447, 1.35127696561955395342231340934, 1.43982470688467799063703792606, 1.89566026897872690270131315116, 1.90746687134987415547704344010, 1.98285334386829952390424308470, 2.13568900946397547578514621308, 2.19456323228185430533644901576, 2.27729605181723478609908773447, 2.65061006043583440151297038900, 2.86622465931618096063060486825, 2.90932818050755766942528050050, 2.95907924462993947802819148264, 2.96643102865031711709677636882, 3.13976128509321437329012400558, 3.16614058657274622918266973248, 3.49848608694700097849450065210, 3.59928833084004307463695668842
Plot not available for L-functions of degree greater than 10.
