| L(s) = 1 | + 18·9-s + 12·11-s + 152·19-s + 120·29-s − 8·31-s − 180·41-s − 44·49-s + 348·59-s + 88·61-s + 320·79-s + 81·81-s + 216·99-s − 888·101-s + 800·109-s − 46·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 356·169-s + 2.73e3·171-s + 173-s + ⋯ |
| L(s) = 1 | + 2·9-s + 1.09·11-s + 8·19-s + 4.13·29-s − 0.258·31-s − 4.39·41-s − 0.897·49-s + 5.89·59-s + 1.44·61-s + 4.05·79-s + 81-s + 2.18·99-s − 8.79·101-s + 7.33·109-s − 0.380·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s − 2.10·169-s + 16·171-s + 0.00578·173-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{16} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{16} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(81.02984487\) |
| \(L(\frac12)\) |
\(\approx\) |
\(81.02984487\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( ( 1 - p^{2} T^{2} + p^{4} T^{4} )^{2} \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 44 T^{2} + 10 p^{2} T^{4} - 147664 T^{6} - 8322221 T^{8} - 147664 p^{4} T^{10} + 10 p^{10} T^{12} + 44 p^{12} T^{14} + p^{16} T^{16} \) |
| 11 | \( ( 1 - 6 T + 7 p T^{2} - 390 T^{3} - 8964 T^{4} - 390 p^{2} T^{5} + 7 p^{5} T^{6} - 6 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 13 | \( 1 + 356 T^{2} + 61930 T^{4} + 2735504 T^{6} - 235619981 T^{8} + 2735504 p^{4} T^{10} + 61930 p^{8} T^{12} + 356 p^{12} T^{14} + p^{16} T^{16} \) |
| 17 | \( ( 1 + 790 T^{2} + 320907 T^{4} + 790 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 19 | \( ( 1 - 2 p T + 1023 T^{2} - 2 p^{3} T^{3} + p^{4} T^{4} )^{4} \) |
| 23 | \( 1 - 580 T^{2} - 7354 p T^{4} + 31401200 T^{6} + 83123216083 T^{8} + 31401200 p^{4} T^{10} - 7354 p^{9} T^{12} - 580 p^{12} T^{14} + p^{16} T^{16} \) |
| 29 | \( ( 1 - 60 T + 2462 T^{2} - 75720 T^{3} + 1894563 T^{4} - 75720 p^{2} T^{5} + 2462 p^{4} T^{6} - 60 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 31 | \( ( 1 + 4 T + 1030 T^{2} - 11744 T^{3} + 89299 T^{4} - 11744 p^{2} T^{5} + 1030 p^{4} T^{6} + 4 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 37 | \( ( 1 - 2542 T^{2} + p^{4} T^{4} )^{4} \) |
| 41 | \( ( 1 + 45 T + 2356 T^{2} + 45 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 43 | \( 1 + 5234 T^{2} + 14173105 T^{4} + 33414112466 T^{6} + 69466635680644 T^{8} + 33414112466 p^{4} T^{10} + 14173105 p^{8} T^{12} + 5234 p^{12} T^{14} + p^{16} T^{16} \) |
| 47 | \( 1 - 1540 T^{2} - 5483702 T^{4} + 2932252400 T^{6} + 26321831389843 T^{8} + 2932252400 p^{4} T^{10} - 5483702 p^{8} T^{12} - 1540 p^{12} T^{14} + p^{16} T^{16} \) |
| 53 | \( ( 1 + 9700 T^{2} + 38880102 T^{4} + 9700 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 59 | \( ( 1 - 174 T + 19397 T^{2} - 1619070 T^{3} + 109595916 T^{4} - 1619070 p^{2} T^{5} + 19397 p^{4} T^{6} - 174 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 61 | \( ( 1 - 44 T - 5450 T^{2} + 2464 T^{3} + 33503299 T^{4} + 2464 p^{2} T^{5} - 5450 p^{4} T^{6} - 44 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 67 | \( 1 + 578 T^{2} - 38358239 T^{4} - 930533182 T^{6} + 1084296014471524 T^{8} - 930533182 p^{4} T^{10} - 38358239 p^{8} T^{12} + 578 p^{12} T^{14} + p^{16} T^{16} \) |
| 71 | \( ( 1 - 9148 T^{2} + 46550598 T^{4} - 9148 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 73 | \( ( 1 - 17498 T^{2} + 130384443 T^{4} - 17498 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 79 | \( ( 1 - 160 T + 7258 T^{2} - 937600 T^{3} + 137709283 T^{4} - 937600 p^{2} T^{5} + 7258 p^{4} T^{6} - 160 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 83 | \( ( 1 - 5666 T^{2} - 15354765 T^{4} - 5666 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 89 | \( ( 1 - 22468 T^{2} + 231780678 T^{4} - 22468 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 97 | \( 1 + 17354 T^{2} + 48852985 T^{4} + 1305884491226 T^{6} + 29160465527134804 T^{8} + 1305884491226 p^{4} T^{10} + 48852985 p^{8} T^{12} + 17354 p^{12} T^{14} + p^{16} T^{16} \) |
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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
−4.02201352482008577387494852335, −4.01448956601388805185120040136, −3.82424112567151843908827656150, −3.62110927523522180547421584666, −3.61986919763359253604136762308, −3.42275294769675509397109598585, −3.24179543834265442627308076118, −3.15132945285688429722265339057, −3.07160776117176220826866322952, −3.04260566198059395781726596570, −2.90989473658137376593641469928, −2.60740280013591063859911202989, −2.40223284948196197446075783499, −2.36958609519045993893076349358, −1.90941508953867006656114900792, −1.81305982315601407449485935726, −1.78108825576746782799495542322, −1.54662508118962560776353672621, −1.22819006667644803088978458605, −1.22571277320524442262000080834, −0.986154442202758948593875809250, −0.848862675417501290509463010043, −0.830536382739507228732134861132, −0.53122330169873913432717344645, −0.48587360948178513112908463987,
0.48587360948178513112908463987, 0.53122330169873913432717344645, 0.830536382739507228732134861132, 0.848862675417501290509463010043, 0.986154442202758948593875809250, 1.22571277320524442262000080834, 1.22819006667644803088978458605, 1.54662508118962560776353672621, 1.78108825576746782799495542322, 1.81305982315601407449485935726, 1.90941508953867006656114900792, 2.36958609519045993893076349358, 2.40223284948196197446075783499, 2.60740280013591063859911202989, 2.90989473658137376593641469928, 3.04260566198059395781726596570, 3.07160776117176220826866322952, 3.15132945285688429722265339057, 3.24179543834265442627308076118, 3.42275294769675509397109598585, 3.61986919763359253604136762308, 3.62110927523522180547421584666, 3.82424112567151843908827656150, 4.01448956601388805185120040136, 4.02201352482008577387494852335
Plot not available for L-functions of degree greater than 10.
