| L(s) = 1 | + 8·13-s + 16·17-s + 16·29-s + 24·37-s + 8·49-s − 48·53-s + 24·61-s − 2·81-s − 32·97-s + 16·101-s + 24·109-s + 16·113-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 32·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
| L(s) = 1 | + 2.21·13-s + 3.88·17-s + 2.97·29-s + 3.94·37-s + 8/7·49-s − 6.59·53-s + 3.07·61-s − 2/9·81-s − 3.24·97-s + 1.59·101-s + 2.29·109-s + 1.50·113-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 + 2.46·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{72} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{72} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.008534865\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.008534865\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( ( 1 + T^{4} )^{2} \) |
| good | 5 | \( ( 1 - p T^{2} )^{4}( 1 + p T^{2} )^{4} \) |
| 7 | \( ( 1 - 4 T^{2} + 22 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 11 | \( 1 + 164 T^{4} + 30886 T^{8} + 164 p^{4} T^{12} + p^{8} T^{16} \) |
| 13 | \( ( 1 - 6 T + p T^{2} )^{4}( 1 + 4 T + p T^{2} )^{4} \) |
| 17 | \( ( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{4} \) |
| 19 | \( 1 - 412 T^{4} + 52198 T^{8} - 412 p^{4} T^{12} + p^{8} T^{16} \) |
| 23 | \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{4} \) |
| 29 | \( ( 1 - 8 T + 32 T^{2} - 216 T^{3} + 1454 T^{4} - 216 p T^{5} + 32 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 31 | \( ( 1 + 4 T^{2} - 74 T^{4} + 4 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 - 12 T + 72 T^{2} - 180 T^{3} - 34 T^{4} - 180 p T^{5} + 72 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 41 | \( ( 1 - 116 T^{2} + 6406 T^{4} - 116 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 43 | \( 1 + 164 T^{4} - 3523674 T^{8} + 164 p^{4} T^{12} + p^{8} T^{16} \) |
| 47 | \( ( 1 + p T^{2} )^{8} \) |
| 53 | \( ( 1 + 24 T + 288 T^{2} + 2760 T^{3} + 22606 T^{4} + 2760 p T^{5} + 288 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 59 | \( ( 1 - 5518 T^{4} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 - 12 T + 72 T^{2} - 468 T^{3} + 2558 T^{4} - 468 p T^{5} + 72 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 67 | \( ( 1 - 8878 T^{4} + p^{4} T^{8} )^{2} \) |
| 71 | \( ( 1 - 188 T^{2} + 17638 T^{4} - 188 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 73 | \( ( 1 - 60 T^{2} + 38 T^{4} - 60 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 79 | \( ( 1 + 260 T^{2} + 28662 T^{4} + 260 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 83 | \( 1 + 13412 T^{4} + 114077158 T^{8} + 13412 p^{4} T^{12} + p^{8} T^{16} \) |
| 89 | \( ( 1 - 16 T + p T^{2} )^{4}( 1 + 16 T + p T^{2} )^{4} \) |
| 97 | \( ( 1 + 8 T + 130 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{4} \) |
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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
−3.97222376987784241290066392641, −3.89909434353405739871051793846, −3.81628896111339860868829726060, −3.52531488446330335189865268430, −3.50190321659178214189698981733, −3.24794731641857081841506584908, −3.24743964823838078418687159156, −3.14896880146235987779955882548, −2.95469358646121661711511745897, −2.94731766489674032885058649383, −2.78478215279511310620328637541, −2.73221561573729485995345977467, −2.38834539746110669562443880735, −2.26498807120745431356380887716, −2.14216328638014757530548573511, −2.01564877535526630651729239743, −1.66537712109712340814262770467, −1.62364611119692392001815848341, −1.34750322721300157694081995435, −1.19279469659792457532055827507, −1.01097261390289622769491005105, −0.985175708008266337069115518338, −0.867897769378288760701957879509, −0.77761900466769032052845230262, −0.06506538176711891099049079425,
0.06506538176711891099049079425, 0.77761900466769032052845230262, 0.867897769378288760701957879509, 0.985175708008266337069115518338, 1.01097261390289622769491005105, 1.19279469659792457532055827507, 1.34750322721300157694081995435, 1.62364611119692392001815848341, 1.66537712109712340814262770467, 2.01564877535526630651729239743, 2.14216328638014757530548573511, 2.26498807120745431356380887716, 2.38834539746110669562443880735, 2.73221561573729485995345977467, 2.78478215279511310620328637541, 2.94731766489674032885058649383, 2.95469358646121661711511745897, 3.14896880146235987779955882548, 3.24743964823838078418687159156, 3.24794731641857081841506584908, 3.50190321659178214189698981733, 3.52531488446330335189865268430, 3.81628896111339860868829726060, 3.89909434353405739871051793846, 3.97222376987784241290066392641
Plot not available for L-functions of degree greater than 10.
