| L(s) = 1 | + 24·9-s + 3·16-s − 20·25-s − 32·59-s + 324·81-s + 44·121-s + 127-s + 131-s + 137-s + 139-s + 72·144-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 + 223-s − 480·225-s + 227-s + ⋯ |
| L(s) = 1 | + 8·9-s + 3/4·16-s − 4·25-s − 4.16·59-s + 36·81-s + 4·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 6·144-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 + 0.0669·223-s − 32·225-s + 0.0663·227-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{8} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{8} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(11.61610348\) |
| \(L(\frac12)\) |
\(\approx\) |
\(11.61610348\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 3 T^{4} + p^{4} T^{8} \) |
| 5 | \( ( 1 + p T^{2} )^{4} \) |
| 11 | \( ( 1 - p T^{2} )^{4} \) |
| good | 3 | \( ( 1 - p T^{2} )^{8} \) |
| 7 | \( ( 1 - 78 T^{4} + p^{4} T^{8} )^{2} \) |
| 13 | \( ( 1 + 162 T^{4} + p^{4} T^{8} )^{2} \) |
| 17 | \( ( 1 + 402 T^{4} + p^{4} T^{8} )^{2} \) |
| 19 | \( ( 1 - p T^{2} )^{8} \) |
| 23 | \( ( 1 + p T^{2} )^{8} \) |
| 29 | \( ( 1 + p T^{2} )^{8} \) |
| 31 | \( ( 1 + 18 T^{2} + p^{2} T^{4} )^{4} \) |
| 37 | \( ( 1 + p T^{2} )^{8} \) |
| 41 | \( ( 1 - p T^{2} )^{8} \) |
| 43 | \( ( 1 + 3522 T^{4} + p^{4} T^{8} )^{2} \) |
| 47 | \( ( 1 + p T^{2} )^{8} \) |
| 53 | \( ( 1 + p T^{2} )^{8} \) |
| 59 | \( ( 1 + 4 T + p T^{2} )^{8} \) |
| 61 | \( ( 1 + p T^{2} )^{8} \) |
| 67 | \( ( 1 - p T^{2} )^{8} \) |
| 71 | \( ( 1 - 8 T + p T^{2} )^{4}( 1 + 8 T + p T^{2} )^{4} \) |
| 73 | \( ( 1 - 10638 T^{4} + p^{4} T^{8} )^{2} \) |
| 79 | \( ( 1 + p T^{2} )^{8} \) |
| 83 | \( ( 1 + 13602 T^{4} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{4} \) |
| 97 | \( ( 1 - p T^{2} )^{8} \) |
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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
−4.75878713840564443509196778712, −4.63986528109313714569756561661, −4.57323542968418820370244060559, −4.56854975919748306827623734219, −4.37235030950362240854056711885, −4.02597956323174448780330672281, −3.94723977293757253136950591216, −3.93284646746835353349176796951, −3.91418779930792673463398559075, −3.88827883951702434038043298902, −3.68880475980048710200765409470, −3.21652829301743973662760598595, −3.20377256278119633203474158549, −3.00352123287499253346728047560, −2.93488902559951309423605711162, −2.25872236264795694304878807226, −2.15156041156792855193469903162, −2.00221807795488904148558950136, −1.92334628374516996205980650557, −1.76684075556930808688493665619, −1.51273901426729815649090301606, −1.37218370888045847559374843920, −1.34942630710739335362661198599, −0.855905429141141288064684164211, −0.60588845843525547709351263525,
0.60588845843525547709351263525, 0.855905429141141288064684164211, 1.34942630710739335362661198599, 1.37218370888045847559374843920, 1.51273901426729815649090301606, 1.76684075556930808688493665619, 1.92334628374516996205980650557, 2.00221807795488904148558950136, 2.15156041156792855193469903162, 2.25872236264795694304878807226, 2.93488902559951309423605711162, 3.00352123287499253346728047560, 3.20377256278119633203474158549, 3.21652829301743973662760598595, 3.68880475980048710200765409470, 3.88827883951702434038043298902, 3.91418779930792673463398559075, 3.93284646746835353349176796951, 3.94723977293757253136950591216, 4.02597956323174448780330672281, 4.37235030950362240854056711885, 4.56854975919748306827623734219, 4.57323542968418820370244060559, 4.63986528109313714569756561661, 4.75878713840564443509196778712
Plot not available for L-functions of degree greater than 10.
