| L(s) = 1 | − 4·4-s + 4·5-s + 4·7-s + 8·11-s + 4·13-s + 12·16-s − 8·19-s − 16·20-s − 12·23-s + 10·25-s − 16·28-s + 4·29-s − 16·31-s + 16·35-s + 4·37-s + 12·41-s + 16·43-s − 32·44-s + 8·49-s − 16·52-s − 4·53-s + 32·55-s + 16·59-s + 4·61-s − 32·64-s + 16·65-s − 8·67-s + ⋯ |
| L(s) = 1 | − 2·4-s + 1.78·5-s + 1.51·7-s + 2.41·11-s + 1.10·13-s + 3·16-s − 1.83·19-s − 3.57·20-s − 2.50·23-s + 2·25-s − 3.02·28-s + 0.742·29-s − 2.87·31-s + 2.70·35-s + 0.657·37-s + 1.87·41-s + 2.43·43-s − 4.82·44-s + 8/7·49-s − 2.21·52-s − 0.549·53-s + 4.31·55-s + 2.08·59-s + 0.512·61-s − 4·64-s + 1.98·65-s − 0.977·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.523153842\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.523153842\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2$$\times$$C_2^2$ | \( ( 1 - 2 T + p T^{2} )^{2}( 1 - 8 T^{2} + p^{2} T^{4} ) \) |
| 7 | $C_2^2$ | \( ( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2$$\times$$C_2^2$ | \( ( 1 - 6 T + p T^{2} )^{2}( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} ) \) |
| 13 | $D_4\times C_2$ | \( 1 - 4 T + 6 T^{2} - 4 T^{3} + 2 T^{4} - 4 p T^{5} + 6 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2^2$ | \( ( 1 - 26 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $D_4\times C_2$ | \( 1 + 8 T + 18 T^{2} - 160 T^{3} - 1246 T^{4} - 160 p T^{5} + 18 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 + 12 T + 72 T^{2} + 300 T^{3} + 1246 T^{4} + 300 p T^{5} + 72 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 4 T + 6 T^{2} + 204 T^{3} - 830 T^{4} + 204 p T^{5} + 6 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 37 | $D_4\times C_2$ | \( 1 - 4 T + 6 T^{2} - 4 T^{3} + 2 T^{4} - 4 p T^{5} + 6 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $D_4\times C_2$ | \( 1 - 12 T + 72 T^{2} - 516 T^{3} + 3694 T^{4} - 516 p T^{5} + 72 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 - 16 T + 162 T^{2} - 1384 T^{3} + 10178 T^{4} - 1384 p T^{5} + 162 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 52 T^{2} + 486 T^{4} - 52 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 + 4 T + 54 T^{2} + 708 T^{3} + 3490 T^{4} + 708 p T^{5} + 54 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 16 T + 114 T^{2} - 696 T^{3} + 4834 T^{4} - 696 p T^{5} + 114 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 4 T + 6 T^{2} - 4 T^{3} + 2 T^{4} - 4 p T^{5} + 6 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 + 8 T + 18 T^{2} - 736 T^{3} - 5854 T^{4} - 736 p T^{5} + 18 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 - 12 T + 72 T^{2} - 876 T^{3} + 10654 T^{4} - 876 p T^{5} + 72 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2^2$ | \( ( 1 - 14 T + 98 T^{2} - 14 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 - 122 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 + 16 T + 114 T^{2} + 792 T^{3} + 6370 T^{4} + 792 p T^{5} + 114 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 + 12 T + 72 T^{2} + 516 T^{3} + 1582 T^{4} + 516 p T^{5} + 72 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 + 20 T + 222 T^{2} + 20 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−8.638994326805856693751953397054, −8.276112455032660916358225904149, −8.105470077520734366138473687265, −8.079326517441950949999712817197, −8.023693898639926356864332301895, −7.07091315322006803043946179214, −7.00672950726704440699684843683, −6.89015648657977199746701754516, −6.34542524003591990203514964658, −6.03131193345928915834674960077, −5.81877684317307320072087705723, −5.72601722511956171202802036461, −5.49396425573329848786589786716, −5.23564127947105950268077066049, −4.61125174880022167858829821708, −4.28166331738497335254708313569, −4.21452862130533192821649416226, −3.94619168554748693802608659141, −3.90956677656570595161448720868, −3.31387965729759058699251710615, −2.56908370604028217141744108294, −2.06312245453306001494040830920, −1.87228195633923182254278238830, −1.32544167964676166163749752552, −0.926477235343415300352946675878,
0.926477235343415300352946675878, 1.32544167964676166163749752552, 1.87228195633923182254278238830, 2.06312245453306001494040830920, 2.56908370604028217141744108294, 3.31387965729759058699251710615, 3.90956677656570595161448720868, 3.94619168554748693802608659141, 4.21452862130533192821649416226, 4.28166331738497335254708313569, 4.61125174880022167858829821708, 5.23564127947105950268077066049, 5.49396425573329848786589786716, 5.72601722511956171202802036461, 5.81877684317307320072087705723, 6.03131193345928915834674960077, 6.34542524003591990203514964658, 6.89015648657977199746701754516, 7.00672950726704440699684843683, 7.07091315322006803043946179214, 8.023693898639926356864332301895, 8.079326517441950949999712817197, 8.105470077520734366138473687265, 8.276112455032660916358225904149, 8.638994326805856693751953397054
