| L(s) = 1 | − 16·7-s + 8·13-s + 32·19-s − 10·25-s + 8·31-s − 136·37-s + 80·43-s + 144·49-s − 40·61-s − 304·67-s + 152·73-s + 200·79-s − 128·91-s − 424·97-s − 112·103-s + 104·109-s + 88·121-s + 127-s + 131-s − 512·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | − 2.28·7-s + 8/13·13-s + 1.68·19-s − 2/5·25-s + 8/31·31-s − 3.67·37-s + 1.86·43-s + 2.93·49-s − 0.655·61-s − 4.53·67-s + 2.08·73-s + 2.53·79-s − 1.40·91-s − 4.37·97-s − 1.08·103-s + 0.954·109-s + 8/11·121-s + 0.00787·127-s + 0.00763·131-s − 3.84·133-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.7939237281\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7939237281\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| good | 7 | $D_{4}$ | \( ( 1 + 8 T + 24 T^{2} + 8 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 11 | $D_4\times C_2$ | \( 1 - 8 p T^{2} + 18258 T^{4} - 8 p^{5} T^{6} + p^{8} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 - 4 T + 252 T^{2} - 4 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 290 T^{2} + p^{4} T^{4} )^{2} \) |
| 19 | $D_{4}$ | \( ( 1 - 16 T + 426 T^{2} - 16 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 1612 T^{2} + 1157478 T^{4} - 1612 p^{4} T^{6} + p^{8} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 700 T^{2} + 707622 T^{4} - 700 p^{4} T^{6} + p^{8} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 - 4 T + 1566 T^{2} - 4 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 37 | $D_{4}$ | \( ( 1 + 68 T + 3084 T^{2} + 68 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 - 640 T^{2} + 3887682 T^{4} - 640 p^{4} T^{6} + p^{8} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 - 40 T + 3738 T^{2} - 40 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 - 340 T^{2} - 372378 T^{4} - 340 p^{4} T^{6} + p^{8} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 1876 T^{2} - 4075194 T^{4} - 1876 p^{4} T^{6} + p^{8} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 280 T^{2} - 9454638 T^{4} - 280 p^{4} T^{6} + p^{8} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 + 20 T + 4302 T^{2} + 20 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 76 T + p^{2} T^{2} )^{4} \) |
| 71 | $D_4\times C_2$ | \( 1 - 13540 T^{2} + 89191302 T^{4} - 13540 p^{4} T^{6} + p^{8} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 - 76 T + 8862 T^{2} - 76 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 - 100 T + 11742 T^{2} - 100 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 21652 T^{2} + 211289478 T^{4} - 21652 p^{4} T^{6} + p^{8} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 14368 T^{2} + 154232898 T^{4} - 14368 p^{4} T^{6} + p^{8} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 + 212 T + 29694 T^{2} + 212 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
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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
−9.129727895260977836775490460475, −9.067260749798397705614813280084, −8.363092511700564856928805809288, −8.343063343726860762457119693968, −8.062887184178291795752613879799, −7.43532395252547953467132751341, −7.24338969161511519672884436281, −7.19075937505196346420280331131, −6.90318336016926118128480497204, −6.46754975996789528168434701776, −6.20578586911776896858749017585, −5.89350633994659501334598119124, −5.77697590846670809773509632836, −5.39094811154569674040040368375, −4.91602170038939271475228231455, −4.78243456412185717338955515537, −4.05955638928157510233236492774, −3.79106351997745208391470731416, −3.52802258223108513047517291618, −3.22548182864190025138127961416, −2.85437919936228308911185311067, −2.50529463217946527320566340079, −1.71370934504568745414697561263, −1.18719959808067443035170333066, −0.30386543257198405174188844239,
0.30386543257198405174188844239, 1.18719959808067443035170333066, 1.71370934504568745414697561263, 2.50529463217946527320566340079, 2.85437919936228308911185311067, 3.22548182864190025138127961416, 3.52802258223108513047517291618, 3.79106351997745208391470731416, 4.05955638928157510233236492774, 4.78243456412185717338955515537, 4.91602170038939271475228231455, 5.39094811154569674040040368375, 5.77697590846670809773509632836, 5.89350633994659501334598119124, 6.20578586911776896858749017585, 6.46754975996789528168434701776, 6.90318336016926118128480497204, 7.19075937505196346420280331131, 7.24338969161511519672884436281, 7.43532395252547953467132751341, 8.062887184178291795752613879799, 8.343063343726860762457119693968, 8.363092511700564856928805809288, 9.067260749798397705614813280084, 9.129727895260977836775490460475
