| 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 − 9·81-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 + ⋯ |
| L(s) = 1 | − 2.28·7-s − 0.727·11-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 − 1/9·81-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.2579002153\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2579002153\) |
| \(L(2)\) |
|
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 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 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
−7.38142771920663094869710280411, −7.27212792957084482973214896096, −7.04538274013922257161587867060, −6.70205471095572788084374991571, −6.39761342182688700971720747017, −6.28524275916549420528355952655, −6.21138990198969338118410791403, −5.84341168544500949344823905561, −5.42125543277861489428358679135, −5.33637529274261670157165206471, −5.28518365645807134350574525238, −4.58914131889427368340486590154, −4.46912229476217038774568087387, −4.20323942650153204725676541384, −3.84122673789808709076965107921, −3.52620287020015905475628944133, −3.22293098509888874249028863338, −3.12400189199110717230912952118, −3.03632361913156934307626177787, −2.10938865789527345851276955352, −2.06966949434371884764869427859, −1.91053696497116419494516166476, −1.35686505998522569282560066798, −0.27080222351857792779891894534, −0.25088929147185379383329443280,
0.25088929147185379383329443280, 0.27080222351857792779891894534, 1.35686505998522569282560066798, 1.91053696497116419494516166476, 2.06966949434371884764869427859, 2.10938865789527345851276955352, 3.03632361913156934307626177787, 3.12400189199110717230912952118, 3.22293098509888874249028863338, 3.52620287020015905475628944133, 3.84122673789808709076965107921, 4.20323942650153204725676541384, 4.46912229476217038774568087387, 4.58914131889427368340486590154, 5.28518365645807134350574525238, 5.33637529274261670157165206471, 5.42125543277861489428358679135, 5.84341168544500949344823905561, 6.21138990198969338118410791403, 6.28524275916549420528355952655, 6.39761342182688700971720747017, 6.70205471095572788084374991571, 7.04538274013922257161587867060, 7.27212792957084482973214896096, 7.38142771920663094869710280411
