| L(s) = 1 | + 4·2-s + 6·4-s − 32·11-s − 15·16-s − 128·22-s + 8·23-s − 8·25-s + 20·29-s − 24·32-s + 8·37-s − 20·43-s − 192·44-s + 32·46-s − 32·50-s + 4·53-s + 80·58-s − 6·64-s + 12·67-s + 48·71-s + 32·74-s − 8·79-s − 80·86-s + 48·92-s − 48·100-s + 16·106-s + 32·107-s − 52·109-s + ⋯ |
| L(s) = 1 | + 2.82·2-s + 3·4-s − 9.64·11-s − 3.75·16-s − 27.2·22-s + 1.66·23-s − 8/5·25-s + 3.71·29-s − 4.24·32-s + 1.31·37-s − 3.04·43-s − 28.9·44-s + 4.71·46-s − 4.52·50-s + 0.549·53-s + 10.5·58-s − 3/4·64-s + 1.46·67-s + 5.69·71-s + 3.71·74-s − 0.900·79-s − 8.62·86-s + 5.00·92-s − 4.79·100-s + 1.55·106-s + 3.09·107-s − 4.98·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{24} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{24} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.817220774\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.817220774\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 - T + T^{2} )^{4} \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( ( 1 + 4 T^{2} + 39 T^{4} + 4 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 11 | \( ( 1 + 4 T + p T^{2} )^{8} \) |
| 13 | \( ( 1 - 8 T^{2} - 105 T^{4} - 8 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 17 | \( ( 1 - 32 T^{2} + 735 T^{4} - 32 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 19 | \( 1 - 36 T^{2} + 625 T^{4} + 1836 T^{6} - 147936 T^{8} + 1836 p^{2} T^{10} + 625 p^{4} T^{12} - 36 p^{6} T^{14} + p^{8} T^{16} \) |
| 23 | \( ( 1 - 2 T - 13 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{4} \) |
| 29 | \( ( 1 - 10 T + 32 T^{2} - 100 T^{3} + 883 T^{4} - 100 p T^{5} + 32 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 31 | \( ( 1 - 32 T^{2} + 63 T^{4} - 32 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 - 4 T - 2 T^{2} + 224 T^{3} - 1637 T^{4} + 224 p T^{5} - 2 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 41 | \( ( 1 - 50 T^{2} + 819 T^{4} - 50 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 43 | \( ( 1 + 10 T + 4 T^{2} + 100 T^{3} + 3067 T^{4} + 100 p T^{5} + 4 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 47 | \( 1 - 124 T^{2} + 7354 T^{4} - 446896 T^{6} + 25507219 T^{8} - 446896 p^{2} T^{10} + 7354 p^{4} T^{12} - 124 p^{6} T^{14} + p^{8} T^{16} \) |
| 53 | \( ( 1 - 2 T + 32 T^{2} + 268 T^{3} - 2237 T^{4} + 268 p T^{5} + 32 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 59 | \( 1 - 160 T^{2} + 13198 T^{4} - 870400 T^{6} + 52578643 T^{8} - 870400 p^{2} T^{10} + 13198 p^{4} T^{12} - 160 p^{6} T^{14} + p^{8} T^{16} \) |
| 61 | \( 1 - 204 T^{2} + 24145 T^{4} - 2045916 T^{6} + 136536864 T^{8} - 2045916 p^{2} T^{10} + 24145 p^{4} T^{12} - 204 p^{6} T^{14} + p^{8} T^{16} \) |
| 67 | \( ( 1 - 6 T - 92 T^{2} + 36 T^{3} + 9483 T^{4} + 36 p T^{5} - 92 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 71 | \( ( 1 - 12 T + 163 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{4} \) |
| 73 | \( 1 - 36 T^{2} + 2074 T^{4} + 411696 T^{6} - 34699341 T^{8} + 411696 p^{2} T^{10} + 2074 p^{4} T^{12} - 36 p^{6} T^{14} + p^{8} T^{16} \) |
| 79 | \( ( 1 + 4 T - 131 T^{2} - 44 T^{3} + 14104 T^{4} - 44 p T^{5} - 131 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 83 | \( 1 - 256 T^{2} + 36334 T^{4} - 3948544 T^{6} + 353820979 T^{8} - 3948544 p^{2} T^{10} + 36334 p^{4} T^{12} - 256 p^{6} T^{14} + p^{8} T^{16} \) |
| 89 | \( ( 1 - 128 T^{2} + 8463 T^{4} - 128 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 97 | \( 1 + 156 T^{2} + 1594 T^{4} + 612144 T^{6} + 199691859 T^{8} + 612144 p^{2} T^{10} + 1594 p^{4} T^{12} + 156 p^{6} T^{14} + p^{8} T^{16} \) |
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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
−3.61112697714350533891411453262, −3.36966543545860379647076597035, −3.28718134838937234330235257830, −3.24034337973634808640918188061, −3.22605811437590871648727462211, −3.22574868168804037214332036483, −3.07947252815689244431156383709, −2.83576080264526470913519780031, −2.81535614753016922363766570043, −2.62469079405461084408852419995, −2.51015281119501595963112494287, −2.49819066563800663626786018762, −2.31428526509118981838689612162, −2.25062201457549848615663319680, −2.11821391935718285606820694945, −2.09118273767723836306348849653, −2.00519502610533766731331240885, −1.63331271094752114612845717653, −1.28997500538607383898207090920, −1.04934126525059008811751989069, −0.903031220471298887667788815760, −0.60327697687399959557287100999, −0.53354422669433327057079258081, −0.25444873805215802247227216601, −0.19815863951540491637027979659,
0.19815863951540491637027979659, 0.25444873805215802247227216601, 0.53354422669433327057079258081, 0.60327697687399959557287100999, 0.903031220471298887667788815760, 1.04934126525059008811751989069, 1.28997500538607383898207090920, 1.63331271094752114612845717653, 2.00519502610533766731331240885, 2.09118273767723836306348849653, 2.11821391935718285606820694945, 2.25062201457549848615663319680, 2.31428526509118981838689612162, 2.49819066563800663626786018762, 2.51015281119501595963112494287, 2.62469079405461084408852419995, 2.81535614753016922363766570043, 2.83576080264526470913519780031, 3.07947252815689244431156383709, 3.22574868168804037214332036483, 3.22605811437590871648727462211, 3.24034337973634808640918188061, 3.28718134838937234330235257830, 3.36966543545860379647076597035, 3.61112697714350533891411453262
Plot not available for L-functions of degree greater than 10.
