| L(s) = 1 | + 4·3-s − 4·7-s + 10·9-s + 4·11-s + 4·19-s − 16·21-s − 12·23-s − 2·25-s + 20·27-s − 8·29-s − 4·31-s + 16·33-s + 16·37-s − 16·41-s + 12·43-s + 10·49-s + 28·53-s + 16·57-s + 28·59-s − 40·63-s − 48·69-s + 4·71-s − 16·73-s − 8·75-s − 16·77-s − 28·79-s + 34·81-s + ⋯ |
| L(s) = 1 | + 2.30·3-s − 1.51·7-s + 10/3·9-s + 1.20·11-s + 0.917·19-s − 3.49·21-s − 2.50·23-s − 2/5·25-s + 3.84·27-s − 1.48·29-s − 0.718·31-s + 2.78·33-s + 2.63·37-s − 2.49·41-s + 1.82·43-s + 10/7·49-s + 3.84·53-s + 2.11·57-s + 3.64·59-s − 5.03·63-s − 5.77·69-s + 0.474·71-s − 1.87·73-s − 0.923·75-s − 1.82·77-s − 3.15·79-s + 34/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 17^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 17^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(8.000041491\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.000041491\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 17 | | \( 1 \) |
| good | 3 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} )( 1 - 2 T^{2} + p^{2} T^{4} ) \) |
| 5 | $D_4\times C_2$ | \( 1 + 2 T^{2} - 16 T^{3} + 2 T^{4} - 16 p T^{5} + 2 p^{2} T^{6} + p^{4} T^{8} \) |
| 7 | $D_4\times C_2$ | \( 1 + 4 T + 6 T^{2} + 4 T^{3} + 2 T^{4} + 4 p T^{5} + 6 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 4 T + 6 T^{2} + 60 T^{3} - 254 T^{4} + 60 p T^{5} + 6 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $D_4\times C_2$ | \( 1 - 4 T^{2} - 170 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 - 4 T + 8 T^{2} - 20 T^{3} - 146 T^{4} - 20 p T^{5} + 8 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 + 12 T + 38 T^{2} - 212 T^{3} - 2110 T^{4} - 212 p T^{5} + 38 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 + 8 T + 18 T^{2} - 280 T^{3} - 2206 T^{4} - 280 p T^{5} + 18 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 + 4 T + 54 T^{2} + 196 T^{3} + 1442 T^{4} + 196 p T^{5} + 54 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 - 16 T + 114 T^{2} - 608 T^{3} + 3426 T^{4} - 608 p T^{5} + 114 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 46 T^{2} + p^{2} T^{4} )( 1 + 16 T + 128 T^{2} + 16 p T^{3} + p^{2} T^{4} ) \) |
| 43 | $C_2^2$ | \( ( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 - 116 T^{2} + 7270 T^{4} - 116 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 28 T + 392 T^{2} - 3780 T^{3} + 29726 T^{4} - 3780 p T^{5} + 392 p^{2} T^{6} - 28 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 28 T + 392 T^{2} - 3948 T^{3} + 33038 T^{4} - 3948 p T^{5} + 392 p^{2} T^{6} - 28 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 + 2 T^{2} - 240 T^{3} + 2 T^{4} - 240 p T^{5} + 2 p^{2} T^{6} + p^{4} T^{8} \) |
| 67 | $C_2^2$ | \( ( 1 + 6 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $D_4\times C_2$ | \( 1 - 4 T + 54 T^{2} - 996 T^{3} + 4642 T^{4} - 996 p T^{5} + 54 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $D_4\times C_2$ | \( 1 + 16 T + 162 T^{2} + 1744 T^{3} + 15938 T^{4} + 1744 p T^{5} + 162 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 + 28 T + 358 T^{2} + 3260 T^{3} + 28354 T^{4} + 3260 p T^{5} + 358 p^{2} T^{6} + 28 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 + 12 T + 72 T^{2} + 1020 T^{3} + 14446 T^{4} + 1020 p T^{5} + 72 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 212 T^{2} + 25030 T^{4} - 212 p^{2} T^{6} + p^{4} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 + 24 T + 162 T^{2} - 1272 T^{3} - 27774 T^{4} - 1272 p T^{5} + 162 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.26276494559398209320894538677, −6.96489382442785167464942746659, −6.79458967137939028869951522502, −6.18936792108327755057227268505, −6.07529031419576391841195703687, −5.99053195781513057818071622461, −5.78850667379390005389453128614, −5.51034112285584003999456078263, −5.40370827595610317572028777335, −4.71209616800838438125353025189, −4.50884136029629088867453918660, −4.28434068437021909115262298450, −4.14181547289417037771295826555, −3.81405264293460123731197426421, −3.64496013986667222881257646968, −3.54862247667540875992954459399, −3.19491335932151178049542690787, −3.07446116846016556189355284672, −2.41285085967089629930997404417, −2.40239540821402329792332158833, −2.20014642751106620357389178940, −1.86402961504234687590079305915, −1.46425238485708285966436335177, −0.901681567585873293426884696161, −0.54507525774588987198239038861,
0.54507525774588987198239038861, 0.901681567585873293426884696161, 1.46425238485708285966436335177, 1.86402961504234687590079305915, 2.20014642751106620357389178940, 2.40239540821402329792332158833, 2.41285085967089629930997404417, 3.07446116846016556189355284672, 3.19491335932151178049542690787, 3.54862247667540875992954459399, 3.64496013986667222881257646968, 3.81405264293460123731197426421, 4.14181547289417037771295826555, 4.28434068437021909115262298450, 4.50884136029629088867453918660, 4.71209616800838438125353025189, 5.40370827595610317572028777335, 5.51034112285584003999456078263, 5.78850667379390005389453128614, 5.99053195781513057818071622461, 6.07529031419576391841195703687, 6.18936792108327755057227268505, 6.79458967137939028869951522502, 6.96489382442785167464942746659, 7.26276494559398209320894538677
