| L(s) = 1 | − 4·3-s − 6·9-s − 48·13-s + 56·19-s − 4·25-s + 76·27-s + 24·31-s + 120·37-s + 192·39-s + 120·43-s − 20·49-s − 224·57-s + 48·61-s + 136·67-s + 104·73-s + 16·75-s + 192·79-s − 109·81-s − 96·93-s − 360·97-s + 168·103-s − 128·109-s − 480·111-s + 288·117-s + 127-s − 480·129-s + 131-s + ⋯ |
| L(s) = 1 | − 4/3·3-s − 2/3·9-s − 3.69·13-s + 2.94·19-s − 0.159·25-s + 2.81·27-s + 0.774·31-s + 3.24·37-s + 4.92·39-s + 2.79·43-s − 0.408·49-s − 3.92·57-s + 0.786·61-s + 2.02·67-s + 1.42·73-s + 0.213·75-s + 2.43·79-s − 1.34·81-s − 1.03·93-s − 3.71·97-s + 1.63·103-s − 1.17·109-s − 4.32·111-s + 2.46·117-s + 0.00787·127-s − 3.72·129-s + 0.00763·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{4} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{4} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.9127033279\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9127033279\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | $C_2$ | \( ( 1 + 2 T + p^{2} T^{2} )^{2} \) |
| 11 | | \( 1 \) |
| good | 5 | $D_4\times C_2$ | \( 1 + 4 T^{2} - 154 T^{4} + 4 p^{4} T^{6} + p^{8} T^{8} \) |
| 7 | $C_2^2$ | \( ( 1 + 10 T^{2} + p^{4} T^{4} )^{2} \) |
| 13 | $D_{4}$ | \( ( 1 + 24 T + 394 T^{2} + 24 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 - 548 T^{2} + 152006 T^{4} - 548 p^{4} T^{6} + p^{8} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 - 28 T + 566 T^{2} - 28 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 1180 T^{2} + 793734 T^{4} - 1180 p^{4} T^{6} + p^{8} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 2948 T^{2} + 3564710 T^{4} - 2948 p^{4} T^{6} + p^{8} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 - 12 T + 1606 T^{2} - 12 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 37 | $D_{4}$ | \( ( 1 - 60 T + 3286 T^{2} - 60 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 - 3236 T^{2} + 7165574 T^{4} - 3236 p^{4} T^{6} + p^{8} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 - 60 T + 4246 T^{2} - 60 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 - 7964 T^{2} + 25546694 T^{4} - 7964 p^{4} T^{6} + p^{8} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 6524 T^{2} + 24696806 T^{4} - 6524 p^{4} T^{6} + p^{8} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 1540 T^{2} - 4368666 T^{4} - 1540 p^{4} T^{6} + p^{8} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 - 24 T + 7498 T^{2} - 24 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 67 | $D_{4}$ | \( ( 1 - 68 T + 4502 T^{2} - 68 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 71 | $D_4\times C_2$ | \( 1 - 8732 T^{2} + 61537286 T^{4} - 8732 p^{4} T^{6} + p^{8} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 - 52 T + 2534 T^{2} - 52 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 - 96 T + 14698 T^{2} - 96 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 + 6652 T^{2} + 94061606 T^{4} + 6652 p^{4} T^{6} + p^{8} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 29732 T^{2} + 345919238 T^{4} - 29732 p^{4} T^{6} + p^{8} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 + 180 T + 25510 T^{2} + 180 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.67321954106046181155866053425, −6.31364940255741901559767926928, −6.00944955729572811683514667035, −5.90082828437272047396148000849, −5.81583475984769075688208710441, −5.29247778266839213975147349093, −5.23302830495874246745878385997, −5.15532517554871888505815174663, −5.05971497262444643011764104497, −4.63846511600787711900601427529, −4.51416017818631607511843890777, −4.34961704537500083362885421203, −3.74080143551533731040441182292, −3.71283216558026262948024996484, −3.43144371402627636956104242156, −2.76473163668253261281351463024, −2.76052787893445532661200775367, −2.60046323885987394240869822801, −2.45253971728421434040494510903, −2.26678845747336034806445902378, −1.57249660656530435743590173415, −0.972006081581363584428177912542, −0.893897100892765392935253719807, −0.66786531240616640392390745471, −0.19529751454920776921089813023,
0.19529751454920776921089813023, 0.66786531240616640392390745471, 0.893897100892765392935253719807, 0.972006081581363584428177912542, 1.57249660656530435743590173415, 2.26678845747336034806445902378, 2.45253971728421434040494510903, 2.60046323885987394240869822801, 2.76052787893445532661200775367, 2.76473163668253261281351463024, 3.43144371402627636956104242156, 3.71283216558026262948024996484, 3.74080143551533731040441182292, 4.34961704537500083362885421203, 4.51416017818631607511843890777, 4.63846511600787711900601427529, 5.05971497262444643011764104497, 5.15532517554871888505815174663, 5.23302830495874246745878385997, 5.29247778266839213975147349093, 5.81583475984769075688208710441, 5.90082828437272047396148000849, 6.00944955729572811683514667035, 6.31364940255741901559767926928, 6.67321954106046181155866053425
