| L(s) = 1 | + 4·4-s + 2·7-s + 2·13-s + 6·16-s + 2·19-s − 25-s + 8·28-s − 4·43-s + 3·49-s + 8·52-s − 4·61-s − 4·67-s + 8·76-s − 2·79-s + 4·91-s + 4·97-s − 4·100-s + 12·112-s + 2·121-s + 127-s + 131-s + 4·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + ⋯ |
| L(s) = 1 | + 4·4-s + 2·7-s + 2·13-s + 6·16-s + 2·19-s − 25-s + 8·28-s − 4·43-s + 3·49-s + 8·52-s − 4·61-s − 4·67-s + 8·76-s − 2·79-s + 4·91-s + 4·97-s − 4·100-s + 12·112-s + 2·121-s + 127-s + 131-s + 4·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{24} \cdot 43^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{24} \cdot 43^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(3.829504938\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.829504938\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 43 | \( ( 1 + T + T^{2} )^{4} \) |
| good | 2 | \( ( 1 - T^{2} + T^{4} )^{4} \) |
| 5 | \( 1 + T^{2} - T^{6} - T^{8} - T^{10} + T^{14} + T^{16} \) |
| 7 | \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )^{2} \) |
| 11 | \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \) |
| 13 | \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )^{2} \) |
| 17 | \( 1 + T^{2} - T^{6} - T^{8} - T^{10} + T^{14} + T^{16} \) |
| 19 | \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )^{2} \) |
| 23 | \( 1 + T^{2} - T^{6} - T^{8} - T^{10} + T^{14} + T^{16} \) |
| 29 | \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \) |
| 31 | \( ( 1 - T^{2} + T^{4} )^{4} \) |
| 37 | \( ( 1 - T^{2} + T^{4} )^{4} \) |
| 41 | \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \) |
| 47 | \( ( 1 - T )^{8}( 1 + T )^{8} \) |
| 53 | \( 1 + T^{2} - T^{6} - T^{8} - T^{10} + T^{14} + T^{16} \) |
| 59 | \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \) |
| 61 | \( ( 1 + T )^{8}( 1 - T + T^{2} )^{4} \) |
| 67 | \( ( 1 + T )^{8}( 1 - T + T^{2} )^{4} \) |
| 71 | \( ( 1 + T^{2} )^{4}( 1 - T^{2} + T^{4} )^{2} \) |
| 73 | \( ( 1 - T^{2} + T^{4} )^{4} \) |
| 79 | \( ( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} )^{2} \) |
| 83 | \( ( 1 + T^{2} )^{4}( 1 - T^{2} + T^{4} )^{2} \) |
| 89 | \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \) |
| 97 | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{4} \) |
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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
−4.43506787798890369932619542900, −4.25020745765000525940001492280, −4.19540062594556646775632826386, −4.19227203711983259711145917073, −3.90839399965450219923337504182, −3.58786312166263280871750927507, −3.42976903320511324576292803656, −3.42358721837135416809936593045, −3.27839130984698103478532574337, −3.23053993401973310180864534127, −3.21557679055777571319286402888, −2.85301223097835147413910501067, −2.74083826748568944068672135663, −2.64725495857878167579861452491, −2.55963581647521991610905846354, −2.36441836502158253006420749801, −2.14002882726016571885559672979, −2.01174394344928704448951503952, −1.85408965094289176994301350777, −1.60500476550645853405808498542, −1.54107757652419307287380007244, −1.50876603173756198050445163025, −1.41155860770899583969212521682, −1.34277626155514618092388836474, −0.867320819377428045466947014368,
0.867320819377428045466947014368, 1.34277626155514618092388836474, 1.41155860770899583969212521682, 1.50876603173756198050445163025, 1.54107757652419307287380007244, 1.60500476550645853405808498542, 1.85408965094289176994301350777, 2.01174394344928704448951503952, 2.14002882726016571885559672979, 2.36441836502158253006420749801, 2.55963581647521991610905846354, 2.64725495857878167579861452491, 2.74083826748568944068672135663, 2.85301223097835147413910501067, 3.21557679055777571319286402888, 3.23053993401973310180864534127, 3.27839130984698103478532574337, 3.42358721837135416809936593045, 3.42976903320511324576292803656, 3.58786312166263280871750927507, 3.90839399965450219923337504182, 4.19227203711983259711145917073, 4.19540062594556646775632826386, 4.25020745765000525940001492280, 4.43506787798890369932619542900
Plot not available for L-functions of degree greater than 10.
