| L(s) = 1 | + 2·4-s + 2·9-s − 16·11-s + 16-s − 24·19-s − 4·29-s − 24·31-s + 4·36-s − 20·41-s − 32·44-s + 24·49-s + 28·61-s − 2·64-s + 32·71-s − 48·76-s − 16·79-s + 19·81-s − 32·99-s + 44·101-s + 16·109-s − 8·116-s + 84·121-s − 48·124-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | + 4-s + 2/3·9-s − 4.82·11-s + 1/4·16-s − 5.50·19-s − 0.742·29-s − 4.31·31-s + 2/3·36-s − 3.12·41-s − 4.82·44-s + 24/7·49-s + 3.58·61-s − 1/4·64-s + 3.79·71-s − 5.50·76-s − 1.80·79-s + 19/9·81-s − 3.21·99-s + 4.37·101-s + 1.53·109-s − 0.742·116-s + 7.63·121-s − 4.31·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{16} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{16} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.04708635555\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.04708635555\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 5 | \( 1 \) |
| 19 | \( ( 1 + 12 T + 67 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| good | 3 | \( ( 1 - T^{2} - 8 T^{4} - p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 7 | \( ( 1 - 12 T^{2} + 106 T^{4} - 12 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 11 | \( ( 1 + 4 T + 19 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{4} \) |
| 13 | \( ( 1 - T^{2} + p^{2} T^{4} )^{2}( 1 + 23 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | \( ( 1 + p T^{2} + p^{2} T^{4} )^{4} \) |
| 23 | \( 1 + 76 T^{2} + 3302 T^{4} + 107616 T^{6} + 2785331 T^{8} + 107616 p^{2} T^{10} + 3302 p^{4} T^{12} + 76 p^{6} T^{14} + p^{8} T^{16} \) |
| 29 | \( ( 1 + 2 T - 48 T^{2} - 12 T^{3} + 1747 T^{4} - 12 p T^{5} - 48 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 31 | \( ( 1 + 6 T + 64 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{4} \) |
| 37 | \( ( 1 - 116 T^{2} + 5850 T^{4} - 116 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 41 | \( ( 1 + 10 T + 21 T^{2} - 30 T^{3} + 460 T^{4} - 30 p T^{5} + 21 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 43 | \( 1 + 44 T^{2} + 1786 T^{4} - 156112 T^{6} - 6975965 T^{8} - 156112 p^{2} T^{10} + 1786 p^{4} T^{12} + 44 p^{6} T^{14} + p^{8} T^{16} \) |
| 47 | \( 1 + 76 T^{2} + 1286 T^{4} + 5472 T^{6} + 2521235 T^{8} + 5472 p^{2} T^{10} + 1286 p^{4} T^{12} + 76 p^{6} T^{14} + p^{8} T^{16} \) |
| 53 | \( 1 - 20 T^{2} - 3526 T^{4} + 33840 T^{6} + 6399395 T^{8} + 33840 p^{2} T^{10} - 3526 p^{4} T^{12} - 20 p^{6} T^{14} + p^{8} T^{16} \) |
| 59 | \( ( 1 - 55 T^{2} - 456 T^{4} - 55 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 - 14 T + 88 T^{2} + 196 T^{3} - 4013 T^{4} + 196 p T^{5} + 88 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 67 | \( 1 + 246 T^{2} + 36521 T^{4} + 3694182 T^{6} + 283952580 T^{8} + 3694182 p^{2} T^{10} + 36521 p^{4} T^{12} + 246 p^{6} T^{14} + p^{8} T^{16} \) |
| 71 | \( ( 1 - 16 T + 78 T^{2} - 576 T^{3} + 8467 T^{4} - 576 p T^{5} + 78 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 73 | \( 1 + 138 T^{2} + 9113 T^{4} - 100326 T^{6} - 31258956 T^{8} - 100326 p^{2} T^{10} + 9113 p^{4} T^{12} + 138 p^{6} T^{14} + p^{8} T^{16} \) |
| 79 | \( ( 1 - 13 T + p T^{2} )^{4}( 1 + 17 T + p T^{2} )^{4} \) |
| 83 | \( ( 1 - 103 T^{2} + p^{2} T^{4} )^{4} \) |
| 89 | \( ( 1 - p T^{2} + p^{2} T^{4} )^{4} \) |
| 97 | \( 1 + 170 T^{2} + 11929 T^{4} - 313990 T^{6} - 72434540 T^{8} - 313990 p^{2} T^{10} + 11929 p^{4} T^{12} + 170 p^{6} T^{14} + p^{8} T^{16} \) |
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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
−4.31883873706271191512374954578, −4.08575858379804046659958144721, −3.93897393542585119773622167173, −3.89523923551544531744140535943, −3.88330662090817655171408159523, −3.70838220268578919356987380659, −3.42248623021053413759725013182, −3.34683733442975236651821018287, −3.30015015598678295047312669984, −3.07404633784183368897651110187, −3.03312080585753167749507277381, −2.65626233163272111340817305261, −2.30946247568206623010226832340, −2.21859957061049400579447179286, −2.19312011771416872208799565533, −2.17042949538327674722604005457, −2.15479086242211216012008716380, −2.14097735135600506508329049775, −2.13684367990245243669139803842, −1.59784405258202611594329857388, −1.31416053113983390625556210273, −1.02680834663582681728263043362, −0.64144370753660313461971927563, −0.27094206238253439348475482884, −0.05791144585847299424523463822,
0.05791144585847299424523463822, 0.27094206238253439348475482884, 0.64144370753660313461971927563, 1.02680834663582681728263043362, 1.31416053113983390625556210273, 1.59784405258202611594329857388, 2.13684367990245243669139803842, 2.14097735135600506508329049775, 2.15479086242211216012008716380, 2.17042949538327674722604005457, 2.19312011771416872208799565533, 2.21859957061049400579447179286, 2.30946247568206623010226832340, 2.65626233163272111340817305261, 3.03312080585753167749507277381, 3.07404633784183368897651110187, 3.30015015598678295047312669984, 3.34683733442975236651821018287, 3.42248623021053413759725013182, 3.70838220268578919356987380659, 3.88330662090817655171408159523, 3.89523923551544531744140535943, 3.93897393542585119773622167173, 4.08575858379804046659958144721, 4.31883873706271191512374954578
Plot not available for L-functions of degree greater than 10.
