| L(s) = 1 | − 8·2-s + 36·4-s − 120·8-s + 12·13-s + 330·16-s − 4·25-s − 96·26-s − 792·32-s − 12·41-s + 12·47-s − 12·49-s + 32·50-s + 432·52-s − 24·59-s + 1.71e3·64-s + 12·71-s + 12·73-s + 96·82-s − 96·94-s + 96·98-s − 144·100-s − 60·101-s − 1.44e3·104-s + 192·118-s − 64·121-s + 127-s − 3.43e3·128-s + ⋯ |
| L(s) = 1 | − 5.65·2-s + 18·4-s − 42.4·8-s + 3.32·13-s + 82.5·16-s − 4/5·25-s − 18.8·26-s − 140.·32-s − 1.87·41-s + 1.75·47-s − 1.71·49-s + 4.52·50-s + 59.9·52-s − 3.12·59-s + 214.5·64-s + 1.42·71-s + 1.40·73-s + 10.6·82-s − 9.90·94-s + 9.69·98-s − 14.3·100-s − 5.97·101-s − 141.·104-s + 17.6·118-s − 5.81·121-s + 0.0887·127-s − 303.·128-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{16} \cdot 23^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{16} \cdot 23^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1865552089\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1865552089\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 + T )^{8} \) |
| 3 | \( 1 \) |
| 23 | \( 1 \) |
| good | 5 | \( 1 + 4 T^{2} + 37 T^{4} + 256 T^{6} + 604 T^{8} + 256 p^{2} T^{10} + 37 p^{4} T^{12} + 4 p^{6} T^{14} + p^{8} T^{16} \) |
| 7 | \( 1 + 12 T^{2} + 184 T^{4} + 1380 T^{6} + 12462 T^{8} + 1380 p^{2} T^{10} + 184 p^{4} T^{12} + 12 p^{6} T^{14} + p^{8} T^{16} \) |
| 11 | \( ( 1 + 16 T^{2} + p^{2} T^{4} )^{4} \) |
| 13 | \( ( 1 - 6 T + 49 T^{2} - 186 T^{3} + 888 T^{4} - 186 p T^{5} + 49 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 17 | \( 1 + 40 T^{2} + 1201 T^{4} + 23296 T^{6} + 449200 T^{8} + 23296 p^{2} T^{10} + 1201 p^{4} T^{12} + 40 p^{6} T^{14} + p^{8} T^{16} \) |
| 19 | \( 1 + 96 T^{2} + 4801 T^{4} + 154344 T^{6} + 3476352 T^{8} + 154344 p^{2} T^{10} + 4801 p^{4} T^{12} + 96 p^{6} T^{14} + p^{8} T^{16} \) |
| 29 | \( ( 1 + 53 T^{2} + 252 T^{3} + 1128 T^{4} + 252 p T^{5} + 53 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 31 | \( ( 1 + 50 T^{2} + p^{2} T^{4} )^{4} \) |
| 37 | \( 1 + 252 T^{2} + 29224 T^{4} + 2026740 T^{6} + 91751982 T^{8} + 2026740 p^{2} T^{10} + 29224 p^{4} T^{12} + 252 p^{6} T^{14} + p^{8} T^{16} \) |
| 41 | \( ( 1 + 6 T + 113 T^{2} + 594 T^{3} + 6288 T^{4} + 594 p T^{5} + 113 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 43 | \( 1 + 180 T^{2} + 15013 T^{4} + 833688 T^{6} + 37994988 T^{8} + 833688 p^{2} T^{10} + 15013 p^{4} T^{12} + 180 p^{6} T^{14} + p^{8} T^{16} \) |
| 47 | \( ( 1 - 6 T + 134 T^{2} - 462 T^{3} + 7626 T^{4} - 462 p T^{5} + 134 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 53 | \( 1 + 292 T^{2} + 41509 T^{4} + 3749776 T^{6} + 235524844 T^{8} + 3749776 p^{2} T^{10} + 41509 p^{4} T^{12} + 292 p^{6} T^{14} + p^{8} T^{16} \) |
| 59 | \( ( 1 + 12 T + 203 T^{2} + 2130 T^{3} + 16998 T^{4} + 2130 p T^{5} + 203 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 61 | \( 1 - 84 T^{2} + 7021 T^{4} - 311976 T^{6} + 27304092 T^{8} - 311976 p^{2} T^{10} + 7021 p^{4} T^{12} - 84 p^{6} T^{14} + p^{8} T^{16} \) |
| 67 | \( 1 + 420 T^{2} + 82309 T^{4} + 9879864 T^{6} + 797308668 T^{8} + 9879864 p^{2} T^{10} + 82309 p^{4} T^{12} + 420 p^{6} T^{14} + p^{8} T^{16} \) |
| 71 | \( ( 1 - 6 T + 38 T^{2} - 534 T^{3} + 11178 T^{4} - 534 p T^{5} + 38 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 73 | \( ( 1 - 6 T + 130 T^{2} - 1584 T^{3} + 9615 T^{4} - 1584 p T^{5} + 130 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 79 | \( 1 + 168 T^{2} + 14476 T^{4} + 631512 T^{6} + 29280102 T^{8} + 631512 p^{2} T^{10} + 14476 p^{4} T^{12} + 168 p^{6} T^{14} + p^{8} T^{16} \) |
| 83 | \( 1 + 580 T^{2} + 153349 T^{4} + 24076696 T^{6} + 2452314988 T^{8} + 24076696 p^{2} T^{10} + 153349 p^{4} T^{12} + 580 p^{6} T^{14} + p^{8} T^{16} \) |
| 89 | \( 1 + 352 T^{2} + 56785 T^{4} + 6152944 T^{6} + 566976832 T^{8} + 6152944 p^{2} T^{10} + 56785 p^{4} T^{12} + 352 p^{6} T^{14} + p^{8} T^{16} \) |
| 97 | \( 1 + 552 T^{2} + 149401 T^{4} + 25431336 T^{6} + 2953446192 T^{8} + 25431336 p^{2} T^{10} + 149401 p^{4} T^{12} + 552 p^{6} T^{14} + p^{8} T^{16} \) |
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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
−3.01212982468615553150601694090, −2.84124660605530791360209117719, −2.72916223728598444116356418232, −2.55642933216099468438077311874, −2.51833698342167837379330249693, −2.40806584639586126538802125475, −2.37482608809075673858517348236, −2.36000474385316135391289043624, −2.25770410425083121766479489296, −1.77934002057206242163037134062, −1.72980816456324687259622417170, −1.64193100055996590284598229811, −1.58843908156041905077205389457, −1.58604207356770858212218078157, −1.54601136079420687308164094913, −1.47765951244142544388607883523, −1.36122914944754477539049669343, −1.13346455843218826347840822650, −0.872361812391284278050693734228, −0.808715456629294683021095985855, −0.69514370198416298178030572049, −0.58498703495991908436640865326, −0.50233873905416147868709245124, −0.15734844204356683653618836233, −0.15161895261114972287139250364,
0.15161895261114972287139250364, 0.15734844204356683653618836233, 0.50233873905416147868709245124, 0.58498703495991908436640865326, 0.69514370198416298178030572049, 0.808715456629294683021095985855, 0.872361812391284278050693734228, 1.13346455843218826347840822650, 1.36122914944754477539049669343, 1.47765951244142544388607883523, 1.54601136079420687308164094913, 1.58604207356770858212218078157, 1.58843908156041905077205389457, 1.64193100055996590284598229811, 1.72980816456324687259622417170, 1.77934002057206242163037134062, 2.25770410425083121766479489296, 2.36000474385316135391289043624, 2.37482608809075673858517348236, 2.40806584639586126538802125475, 2.51833698342167837379330249693, 2.55642933216099468438077311874, 2.72916223728598444116356418232, 2.84124660605530791360209117719, 3.01212982468615553150601694090
Plot not available for L-functions of degree greater than 10.
