| L(s) = 1 | + 16·7-s − 8·11-s + 16·17-s + 48·23-s + 56·31-s + 32·37-s − 8·41-s − 128·43-s − 80·47-s + 128·49-s − 32·53-s − 120·61-s + 96·67-s − 32·71-s − 256·73-s − 128·77-s − 9·81-s + 160·83-s + 160·97-s + 464·101-s − 336·103-s − 96·107-s + 400·113-s + 256·119-s − 252·121-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | + 16/7·7-s − 0.727·11-s + 0.941·17-s + 2.08·23-s + 1.80·31-s + 0.864·37-s − 0.195·41-s − 2.97·43-s − 1.70·47-s + 2.61·49-s − 0.603·53-s − 1.96·61-s + 1.43·67-s − 0.450·71-s − 3.50·73-s − 1.66·77-s − 1/9·81-s + 1.92·83-s + 1.64·97-s + 4.59·101-s − 3.26·103-s − 0.897·107-s + 3.53·113-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
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(4.544684022\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.544684022\) |
| \(L(2)\) |
|
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 | $D_4\times C_2$ | \( 1 - 16 T + 128 T^{2} - 1104 T^{3} + 9122 T^{4} - 1104 p^{2} T^{5} + 128 p^{4} T^{6} - 16 p^{6} T^{7} + p^{8} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 + 4 T + 150 T^{2} + 4 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 13 | $C_2^3$ | \( 1 + 49154 T^{4} + p^{8} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 16 T + 128 T^{2} - 4368 T^{3} + 148802 T^{4} - 4368 p^{2} T^{5} + 128 p^{4} T^{6} - 16 p^{6} T^{7} + p^{8} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 - 284 T^{2} - 20250 T^{4} - 284 p^{4} T^{6} + p^{8} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 48 T + 1152 T^{2} - 30000 T^{3} + 772034 T^{4} - 30000 p^{2} T^{5} + 1152 p^{4} T^{6} - 48 p^{6} T^{7} + p^{8} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 2084 T^{2} + 2107110 T^{4} - 2084 p^{4} T^{6} + p^{8} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 - 28 T + 582 T^{2} - 28 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 32 T + 512 T^{2} - 29088 T^{3} + 1440962 T^{4} - 29088 p^{2} T^{5} + 512 p^{4} T^{6} - 32 p^{6} T^{7} + p^{8} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + 4 T - 90 T^{2} + 4 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 + 128 T + 8192 T^{2} + 474240 T^{3} + 24009314 T^{4} + 474240 p^{2} T^{5} + 8192 p^{4} T^{6} + 128 p^{6} T^{7} + p^{8} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 + 80 T + 3200 T^{2} + 144720 T^{3} + 6384962 T^{4} + 144720 p^{2} T^{5} + 3200 p^{4} T^{6} + 80 p^{6} T^{7} + p^{8} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 + 32 T + 512 T^{2} - 30432 T^{3} - 12328798 T^{4} - 30432 p^{2} T^{5} + 512 p^{4} T^{6} + 32 p^{6} T^{7} + p^{8} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 6236 T^{2} + 20094246 T^{4} - 6236 p^{4} T^{6} + p^{8} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 + 60 T + 2198 T^{2} + 60 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 96 T + 4608 T^{2} + 16032 T^{3} - 21622558 T^{4} + 16032 p^{2} T^{5} + 4608 p^{4} T^{6} - 96 p^{6} T^{7} + p^{8} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 16 T + 6690 T^{2} + 16 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 + 256 T + 32768 T^{2} + 3350784 T^{3} + 282426242 T^{4} + 3350784 p^{2} T^{5} + 32768 p^{4} T^{6} + 256 p^{6} T^{7} + p^{8} T^{8} \) |
| 79 | $C_2^2$ | \( ( 1 - 8126 T^{2} + p^{4} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 160 T + 12800 T^{2} - 992160 T^{3} + 76431458 T^{4} - 992160 p^{2} T^{5} + 12800 p^{4} T^{6} - 160 p^{6} T^{7} + p^{8} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 30716 T^{2} + 361199046 T^{4} - 30716 p^{4} T^{6} + p^{8} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 - 160 T + 12800 T^{2} + 755040 T^{3} - 155062462 T^{4} + 755040 p^{2} T^{5} + 12800 p^{4} T^{6} - 160 p^{6} T^{7} + p^{8} T^{8} \) |
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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
−7.42089959709615069302506751530, −7.36418124452009385343823846106, −7.05055806518416906286983240304, −6.72905284043207709135304064846, −6.69268124586624776363755588917, −6.02355034488432654737957409021, −6.02058816131976875981437361698, −5.97321866064334385308325736714, −5.29685772993858364616230226141, −5.20051466892227386477345590114, −4.81356121178834796091830035345, −4.80732958080489923024595445545, −4.71830702311499667084920867076, −4.50092828097010976406387144674, −3.89978934635343821303500583008, −3.59111883289716656876763108626, −3.23374805911527931384726727681, −3.03933528362430214789222169546, −2.78482390030949241454414374257, −2.38143540352842597229211449526, −1.78582459790700563048403747824, −1.73008981187764118304115967142, −1.22985635501706034201451394094, −1.02634162241116836353464960090, −0.36054021523354560626617758928,
0.36054021523354560626617758928, 1.02634162241116836353464960090, 1.22985635501706034201451394094, 1.73008981187764118304115967142, 1.78582459790700563048403747824, 2.38143540352842597229211449526, 2.78482390030949241454414374257, 3.03933528362430214789222169546, 3.23374805911527931384726727681, 3.59111883289716656876763108626, 3.89978934635343821303500583008, 4.50092828097010976406387144674, 4.71830702311499667084920867076, 4.80732958080489923024595445545, 4.81356121178834796091830035345, 5.20051466892227386477345590114, 5.29685772993858364616230226141, 5.97321866064334385308325736714, 6.02058816131976875981437361698, 6.02355034488432654737957409021, 6.69268124586624776363755588917, 6.72905284043207709135304064846, 7.05055806518416906286983240304, 7.36418124452009385343823846106, 7.42089959709615069302506751530
