| L(s) = 1 | + 5·4-s + 5·9-s − 2·11-s + 12·16-s − 10·19-s + 20·29-s + 8·31-s + 25·36-s + 8·41-s − 10·44-s + 5·49-s + 20·59-s − 22·61-s + 15·64-s + 18·71-s − 50·76-s + 12·81-s − 10·89-s − 10·99-s + 48·101-s − 50·109-s + 100·116-s − 39·121-s + 40·124-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | + 5/2·4-s + 5/3·9-s − 0.603·11-s + 3·16-s − 2.29·19-s + 3.71·29-s + 1.43·31-s + 25/6·36-s + 1.24·41-s − 1.50·44-s + 5/7·49-s + 2.60·59-s − 2.81·61-s + 15/8·64-s + 2.13·71-s − 5.73·76-s + 4/3·81-s − 1.05·89-s − 1.00·99-s + 4.77·101-s − 4.78·109-s + 9.28·116-s − 3.54·121-s + 3.59·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.443703568\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.443703568\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | | \( 1 \) |
| good | 2 | $D_4\times C_2$ | \( 1 - 5 T^{2} + 13 T^{4} - 5 p^{2} T^{6} + p^{4} T^{8} \) |
| 3 | $D_4\times C_2$ | \( 1 - 5 T^{2} + 13 T^{4} - 5 p^{2} T^{6} + p^{4} T^{8} \) |
| 7 | $D_4\times C_2$ | \( 1 - 5 T^{2} + 93 T^{4} - 5 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $C_4$ | \( ( 1 + T + 21 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 - 10 T^{2} + 283 T^{4} - 10 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 10 T^{2} - 117 T^{4} - 10 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 + 5 T + 33 T^{2} + 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 65 T^{2} + 2013 T^{4} - 65 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 - 10 T + 78 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 37 | $C_2^2$ | \( ( 1 - 65 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $D_{4}$ | \( ( 1 - 4 T + 66 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 100 T^{2} + 4918 T^{4} - 100 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 + 35 T^{2} + 4173 T^{4} + 35 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 140 T^{2} + 9238 T^{4} - 140 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 - 10 T + 63 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 + 11 T + 141 T^{2} + 11 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 130 T^{2} + 9283 T^{4} - 130 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 9 T + 101 T^{2} - 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 245 T^{2} + 25113 T^{4} - 245 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $C_2^2$ | \( ( 1 + 113 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 140 T^{2} + 13558 T^{4} - 140 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 + 5 T + 183 T^{2} + 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 205 T^{2} + 28873 T^{4} - 205 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−7.75039710103851127088159025185, −7.14770745088067745770262634124, −7.11799231622443784892877022722, −6.83152524706206485243017744116, −6.75245633292878567723592975463, −6.49586029220592734135953192725, −6.41308904945353981472817482865, −6.17570014697971685147740348726, −5.89941183800592003666777930235, −5.45087009410928141423052175192, −5.39683920557520008078437888127, −4.77772527176458139093983580739, −4.53751093605098247616197553533, −4.52854130422690155270846071816, −4.27261306585304896282519418034, −3.76022017568674038178614848138, −3.66136201513345715004245735514, −2.90076784701904537406185579134, −2.80940249888711560477957393448, −2.66641329302321302120501466047, −2.32523280144163673300589771234, −1.92386327292494810194899103705, −1.76722983934767260449883778584, −1.06554621277824203713935754396, −0.842135822189655507526748284758,
0.842135822189655507526748284758, 1.06554621277824203713935754396, 1.76722983934767260449883778584, 1.92386327292494810194899103705, 2.32523280144163673300589771234, 2.66641329302321302120501466047, 2.80940249888711560477957393448, 2.90076784701904537406185579134, 3.66136201513345715004245735514, 3.76022017568674038178614848138, 4.27261306585304896282519418034, 4.52854130422690155270846071816, 4.53751093605098247616197553533, 4.77772527176458139093983580739, 5.39683920557520008078437888127, 5.45087009410928141423052175192, 5.89941183800592003666777930235, 6.17570014697971685147740348726, 6.41308904945353981472817482865, 6.49586029220592734135953192725, 6.75245633292878567723592975463, 6.83152524706206485243017744116, 7.11799231622443784892877022722, 7.14770745088067745770262634124, 7.75039710103851127088159025185
