| L(s) = 1 | + 8·11-s + 8·37-s − 48·49-s − 16·59-s − 24·61-s − 72·83-s − 64·97-s + 80·107-s − 48·109-s + 32·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯ |
| L(s) = 1 | + 2.41·11-s + 1.31·37-s − 6.85·49-s − 2.08·59-s − 3.07·61-s − 7.90·83-s − 6.49·97-s + 7.73·107-s − 4.59·109-s + 2.90·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{64} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{64} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.766016914\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.766016914\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 4 T^{4} - 474 T^{8} - 4 p^{4} T^{12} + p^{8} T^{16} \) |
| 7 | \( ( 1 + 12 T^{2} + p^{2} T^{4} )^{4} \) |
| 11 | \( ( 1 - 4 T + 8 T^{2} - 28 T^{3} + 82 T^{4} - 28 p T^{5} + 8 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 13 | \( ( 1 + 62 T^{4} + p^{4} T^{8} )^{2} \) |
| 17 | \( ( 1 - 40 T^{2} + 786 T^{4} - 40 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 19 | \( 1 + 292 T^{4} - 25242 T^{8} + 292 p^{4} T^{12} + p^{8} T^{16} \) |
| 23 | \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{4} \) |
| 29 | \( 1 - 964 T^{4} + 566886 T^{8} - 964 p^{4} T^{12} + p^{8} T^{16} \) |
| 31 | \( ( 1 - 40 T^{2} + 594 T^{4} - 40 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 - 4 T + 8 T^{2} - 60 T^{3} - 34 T^{4} - 60 p T^{5} + 8 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 41 | \( ( 1 + 40 T^{2} + 2034 T^{4} + 40 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 43 | \( 1 - 3548 T^{4} + 6433446 T^{8} - 3548 p^{4} T^{12} + p^{8} T^{16} \) |
| 47 | \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{4} \) |
| 53 | \( 1 - 4868 T^{4} + 16478118 T^{8} - 4868 p^{4} T^{12} + p^{8} T^{16} \) |
| 59 | \( ( 1 + 8 T + 32 T^{2} - 232 T^{3} - 6062 T^{4} - 232 p T^{5} + 32 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{4} \) |
| 67 | \( 1 + 10276 T^{4} + 56007654 T^{8} + 10276 p^{4} T^{12} + p^{8} T^{16} \) |
| 71 | \( ( 1 + 54 T^{2} + p^{2} T^{4} )^{4} \) |
| 73 | \( ( 1 - 188 T^{2} + 18726 T^{4} - 188 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 79 | \( ( 1 - 168 T^{2} + 14738 T^{4} - 168 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 83 | \( ( 1 + 36 T + 648 T^{2} + 8604 T^{3} + 89906 T^{4} + 8604 p T^{5} + 648 p^{2} T^{6} + 36 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 - 80 T^{2} + 15714 T^{4} - 80 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 97 | \( ( 1 + 16 T + 150 T^{2} + 16 p T^{3} + p^{2} T^{4} )^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.69562016608128407554883561247, −3.64844705079875142732115220974, −3.59427615935889594969898649471, −3.33539397811765065158188884433, −3.25289202111295630172314314666, −3.07439284517616794696010746991, −2.99401130541178182733400973168, −2.98361620450611989669930216169, −2.83598452229769350730177130293, −2.71723357658490686711052761215, −2.64164727623675172840482702375, −2.54730588489390712946377078666, −2.13883241054872638421001066532, −2.05601798227476941453800155263, −1.79385424372066582129563774812, −1.57455437724618792499478798722, −1.53449666912448938945438866669, −1.53410337432609390943695285265, −1.52528528022211688623747116455, −1.38206872757384166715666108205, −1.24061885909313295139051244733, −0.75693794620262642423602221787, −0.64846542355813443282390295689, −0.21331046711891688065956724056, −0.20156230752001238148872033334,
0.20156230752001238148872033334, 0.21331046711891688065956724056, 0.64846542355813443282390295689, 0.75693794620262642423602221787, 1.24061885909313295139051244733, 1.38206872757384166715666108205, 1.52528528022211688623747116455, 1.53410337432609390943695285265, 1.53449666912448938945438866669, 1.57455437724618792499478798722, 1.79385424372066582129563774812, 2.05601798227476941453800155263, 2.13883241054872638421001066532, 2.54730588489390712946377078666, 2.64164727623675172840482702375, 2.71723357658490686711052761215, 2.83598452229769350730177130293, 2.98361620450611989669930216169, 2.99401130541178182733400973168, 3.07439284517616794696010746991, 3.25289202111295630172314314666, 3.33539397811765065158188884433, 3.59427615935889594969898649471, 3.64844705079875142732115220974, 3.69562016608128407554883561247
Plot not available for L-functions of degree greater than 10.
