| L(s) = 1 | − 16·7-s + 8·11-s + 16·17-s + 48·23-s − 124·31-s − 32·37-s − 112·41-s − 112·43-s + 160·47-s + 128·49-s + 208·53-s + 300·61-s + 144·67-s + 272·71-s − 224·73-s − 128·77-s + 160·83-s + 320·97-s − 224·101-s + 96·103-s + 144·107-s − 320·113-s − 256·119-s − 252·121-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | − 2.28·7-s + 8/11·11-s + 0.941·17-s + 2.08·23-s − 4·31-s − 0.864·37-s − 2.73·41-s − 2.60·43-s + 3.40·47-s + 2.61·49-s + 3.92·53-s + 4.91·61-s + 2.14·67-s + 3.83·71-s − 3.06·73-s − 1.66·77-s + 1.92·83-s + 3.29·97-s − 2.21·101-s + 0.932·103-s + 1.34·107-s − 2.83·113-s − 2.15·119-s − 2.08·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.194307386\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.194307386\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| good | 7 | $D_4\times C_2$ | \( 1 + 16 T + 128 T^{2} + 864 T^{3} + 5807 T^{4} + 864 p^{2} T^{5} + 128 p^{4} T^{6} + 16 p^{6} T^{7} + p^{8} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 - 4 T + 150 T^{2} - 4 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 13 | $C_2^3$ | \( 1 + 39599 T^{4} + p^{8} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 16 T + 128 T^{2} - 3408 T^{3} + 84962 T^{4} - 3408 p^{2} T^{5} + 128 p^{4} T^{6} - 16 p^{6} T^{7} + p^{8} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 + 286 T^{2} + 277635 T^{4} + 286 p^{4} T^{6} + p^{8} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 48 T + 1152 T^{2} - 1680 p T^{3} + 2306 p^{2} T^{4} - 1680 p^{3} T^{5} + 1152 p^{4} T^{6} - 48 p^{6} T^{7} + p^{8} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 1244 T^{2} + 1124070 T^{4} - 1244 p^{4} T^{6} + p^{8} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 + 2 p T + 2787 T^{2} + 2 p^{3} T^{3} + p^{4} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 + 32 T + 512 T^{2} + 46368 T^{3} + 4192802 T^{4} + 46368 p^{2} T^{5} + 512 p^{4} T^{6} + 32 p^{6} T^{7} + p^{8} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + 56 T + 4050 T^{2} + 56 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 + 112 T + 6272 T^{2} + 366240 T^{3} + 19366559 T^{4} + 366240 p^{2} T^{5} + 6272 p^{4} T^{6} + 112 p^{6} T^{7} + p^{8} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 160 T + 12800 T^{2} - 817440 T^{3} + 43793762 T^{4} - 817440 p^{2} T^{5} + 12800 p^{4} T^{6} - 160 p^{6} T^{7} + p^{8} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 208 T + 21632 T^{2} - 1699152 T^{3} + 104735522 T^{4} - 1699152 p^{2} T^{5} + 21632 p^{4} T^{6} - 208 p^{6} T^{7} + p^{8} T^{8} \) |
| 59 | $C_2^2$ | \( ( 1 - 238 T^{2} + p^{4} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 - 150 T + 12683 T^{2} - 150 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 144 T + 10368 T^{2} - 863712 T^{3} + 69674927 T^{4} - 863712 p^{2} T^{5} + 10368 p^{4} T^{6} - 144 p^{6} T^{7} + p^{8} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 136 T + 14610 T^{2} - 136 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 + 224 T + 25088 T^{2} + 2071776 T^{3} + 155721602 T^{4} + 2071776 p^{2} T^{5} + 25088 p^{4} T^{6} + 224 p^{6} T^{7} + p^{8} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 - 5692 T^{2} + 84617478 T^{4} - 5692 p^{4} T^{6} + p^{8} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 - 160 T + 12800 T^{2} - 1520160 T^{3} + 173715458 T^{4} - 1520160 p^{2} T^{5} + 12800 p^{4} T^{6} - 160 p^{6} T^{7} + p^{8} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 + 8284 T^{2} + 112535046 T^{4} + 8284 p^{4} T^{6} + p^{8} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 - 320 T + 51200 T^{2} - 6760320 T^{3} + 755327663 T^{4} - 6760320 p^{2} T^{5} + 51200 p^{4} T^{6} - 320 p^{6} T^{7} + p^{8} T^{8} \) |
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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
−6.72355911557401911320679707782, −6.12954373238104744816635625840, −5.86536083433017908167701274854, −5.85016144607590765239107067947, −5.48363138661119152388451439520, −5.29633881258149134366329058321, −5.17752803609039957936689399275, −5.13616608123869664978805160793, −4.92678824238729438810122872922, −4.08580922021335705783382740495, −4.04952563558638780087885239675, −3.99473963168527506903874379650, −3.65443841847620653974770641023, −3.50333729506415973106870876690, −3.33629825804808915535068965898, −3.13203376235943205349518852518, −2.92460517439856588655924208776, −2.40024426687312761044545813149, −2.03005754272346242522575066208, −1.94870855458313528813189251801, −1.93993741195009694329035882492, −0.905274722304856252344163841354, −0.905018552013800379320516530146, −0.74043426943753639741081455135, −0.21989879912165033152881117375,
0.21989879912165033152881117375, 0.74043426943753639741081455135, 0.905018552013800379320516530146, 0.905274722304856252344163841354, 1.93993741195009694329035882492, 1.94870855458313528813189251801, 2.03005754272346242522575066208, 2.40024426687312761044545813149, 2.92460517439856588655924208776, 3.13203376235943205349518852518, 3.33629825804808915535068965898, 3.50333729506415973106870876690, 3.65443841847620653974770641023, 3.99473963168527506903874379650, 4.04952563558638780087885239675, 4.08580922021335705783382740495, 4.92678824238729438810122872922, 5.13616608123869664978805160793, 5.17752803609039957936689399275, 5.29633881258149134366329058321, 5.48363138661119152388451439520, 5.85016144607590765239107067947, 5.86536083433017908167701274854, 6.12954373238104744816635625840, 6.72355911557401911320679707782
