| L(s) = 1 | − 3-s + 3·5-s + 7·7-s − 2·9-s + 13-s − 3·15-s + 17-s − 7·19-s − 7·21-s + 8·23-s + 10·25-s + 5·29-s − 5·31-s + 21·35-s − 9·37-s − 39-s + 5·41-s − 6·45-s − 5·47-s + 22·49-s − 51-s + 11·53-s + 7·57-s + 17·59-s + 61-s − 14·63-s + 3·65-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.34·5-s + 2.64·7-s − 2/3·9-s + 0.277·13-s − 0.774·15-s + 0.242·17-s − 1.60·19-s − 1.52·21-s + 1.66·23-s + 2·25-s + 0.928·29-s − 0.898·31-s + 3.54·35-s − 1.47·37-s − 0.160·39-s + 0.780·41-s − 0.894·45-s − 0.729·47-s + 22/7·49-s − 0.140·51-s + 1.51·53-s + 0.927·57-s + 2.21·59-s + 0.128·61-s − 1.76·63-s + 0.372·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.195857080\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.195857080\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 11 | | \( 1 \) |
| good | 3 | $C_2^2:C_4$ | \( 1 + T + p T^{2} + 5 T^{3} + 16 T^{4} + 5 p T^{5} + p^{3} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) |
| 5 | $C_2^2:C_4$ | \( 1 - 3 T - T^{2} + 3 T^{3} + 16 T^{4} + 3 p T^{5} - p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 7 | $C_2^2:C_4$ | \( 1 - p T + 27 T^{2} - 95 T^{3} + 296 T^{4} - 95 p T^{5} + 27 p^{2} T^{6} - p^{4} T^{7} + p^{4} T^{8} \) |
| 13 | $C_2^2:C_4$ | \( 1 - T + 3 T^{2} + 25 T^{3} + 56 T^{4} + 25 p T^{5} + 3 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2^2:C_4$ | \( 1 - T - T^{2} + 53 T^{3} + 104 T^{4} + 53 p T^{5} - p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_2^2:C_4$ | \( 1 + 7 T + 15 T^{2} + 107 T^{3} + 824 T^{4} + 107 p T^{5} + 15 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $D_{4}$ | \( ( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2:C_4$ | \( 1 - 5 T + 31 T^{2} - 115 T^{3} + 96 T^{4} - 115 p T^{5} + 31 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_2^2:C_4$ | \( 1 + 5 T + 9 T^{2} + 205 T^{3} + 1916 T^{4} + 205 p T^{5} + 9 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_2^2:C_4$ | \( 1 + 9 T - T^{2} - 57 T^{3} + 784 T^{4} - 57 p T^{5} - p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2^2:C_4$ | \( 1 - 5 T + 19 T^{2} + 65 T^{3} - 624 T^{4} + 65 p T^{5} + 19 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 47 | $C_2^2:C_4$ | \( 1 + 5 T - 37 T^{2} - 5 p T^{3} + 824 T^{4} - 5 p^{2} T^{5} - 37 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2^2:C_4$ | \( 1 - 11 T + 43 T^{2} - 565 T^{3} + 6936 T^{4} - 565 p T^{5} + 43 p^{2} T^{6} - 11 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_2^2:C_4$ | \( 1 - 17 T + 55 T^{2} + 443 T^{3} - 4776 T^{4} + 443 p T^{5} + 55 p^{2} T^{6} - 17 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2^2:C_4$ | \( 1 - T - 45 T^{2} + 361 T^{3} + 2744 T^{4} + 361 p T^{5} - 45 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_{4}$ | \( ( 1 - 8 T + 70 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2:C_4$ | \( 1 - 5 T - 11 T^{2} + 515 T^{3} - 1164 T^{4} + 515 p T^{5} - 11 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2^2:C_4$ | \( 1 + 19 T + 63 T^{2} - 1015 T^{3} - 13384 T^{4} - 1015 p T^{5} + 63 p^{2} T^{6} + 19 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_2^2:C_4$ | \( 1 + 3 T + 65 T^{2} + 123 T^{3} + 3244 T^{4} + 123 p T^{5} + 65 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_2^2:C_4$ | \( 1 - 31 T + 553 T^{2} - 7355 T^{3} + 75636 T^{4} - 7355 p T^{5} + 553 p^{2} T^{6} - 31 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 + 8 T + 174 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2:C_4$ | \( 1 - 27 T + 227 T^{2} - 585 T^{3} + 256 T^{4} - 585 p T^{5} + 227 p^{2} T^{6} - 27 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−7.971831545909616965577349615314, −7.65778654935867587374017488855, −7.41382281644129266504481000223, −7.27705689039898760396213152032, −6.99020009699597258661366239713, −6.67302088731431645896803212900, −6.28391953474641100465568500152, −6.17989366762387056998513518873, −6.09398667100770180032193753771, −5.59140007141966739848062126577, −5.21724816927639085055682174393, −5.21637355879491133525905772197, −5.04805522510604442264574268995, −4.74396113408630387427084492874, −4.64962761394510519745492396447, −4.07922507356562192692798074882, −3.69800636310515816265694489778, −3.63780121958315386093209429462, −2.85788534075170629109580866644, −2.80521227960194544032912365701, −2.05828146868527219068607943632, −2.03054815940115123780382073172, −1.87018962385584864665881362706, −1.00954153118647406358483651940, −0.890890576571586696560683311209,
0.890890576571586696560683311209, 1.00954153118647406358483651940, 1.87018962385584864665881362706, 2.03054815940115123780382073172, 2.05828146868527219068607943632, 2.80521227960194544032912365701, 2.85788534075170629109580866644, 3.63780121958315386093209429462, 3.69800636310515816265694489778, 4.07922507356562192692798074882, 4.64962761394510519745492396447, 4.74396113408630387427084492874, 5.04805522510604442264574268995, 5.21637355879491133525905772197, 5.21724816927639085055682174393, 5.59140007141966739848062126577, 6.09398667100770180032193753771, 6.17989366762387056998513518873, 6.28391953474641100465568500152, 6.67302088731431645896803212900, 6.99020009699597258661366239713, 7.27705689039898760396213152032, 7.41382281644129266504481000223, 7.65778654935867587374017488855, 7.971831545909616965577349615314
