| L(s) = 1 | + 5·2-s + 3-s + 12·4-s − 3·5-s + 5·6-s − 7-s + 20·8-s − 15·10-s + 11-s + 12·12-s + 4·13-s − 5·14-s − 3·15-s + 30·16-s + 6·17-s − 7·19-s − 36·20-s − 21-s + 5·22-s + 22·23-s + 20·24-s + 10·25-s + 20·26-s − 12·28-s + 6·29-s − 15·30-s − 15·31-s + ⋯ |
| L(s) = 1 | + 3.53·2-s + 0.577·3-s + 6·4-s − 1.34·5-s + 2.04·6-s − 0.377·7-s + 7.07·8-s − 4.74·10-s + 0.301·11-s + 3.46·12-s + 1.10·13-s − 1.33·14-s − 0.774·15-s + 15/2·16-s + 1.45·17-s − 1.60·19-s − 8.04·20-s − 0.218·21-s + 1.06·22-s + 4.58·23-s + 4.08·24-s + 2·25-s + 3.92·26-s − 2.26·28-s + 1.11·29-s − 2.73·30-s − 2.69·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 7^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 7^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(11.80287000\) |
| \(L(\frac12)\) |
\(\approx\) |
\(11.80287000\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_4$ | \( 1 - T + T^{2} - T^{3} + T^{4} \) |
| 7 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 11 | $C_4$ | \( 1 - T + 21 T^{2} - p T^{3} + p^{2} T^{4} \) |
| good | 2 | $C_4\times C_2$ | \( 1 - 5 T + 13 T^{2} - 25 T^{3} + 39 T^{4} - 25 p T^{5} + 13 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 5 | $C_2^2:C_4$ | \( 1 + 3 T - T^{2} - 3 T^{3} + 16 T^{4} - 3 p T^{5} - p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $C_2^2:C_4$ | \( 1 - 4 T + 3 T^{2} - 50 T^{3} + 341 T^{4} - 50 p T^{5} + 3 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2^2:C_4$ | \( 1 - 6 T + 19 T^{2} - 132 T^{3} + 829 T^{4} - 132 p T^{5} + 19 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 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 | $D_{4}$ | \( ( 1 - 11 T + 75 T^{2} - 11 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $C_4\times C_2$ | \( 1 - 6 T + 7 T^{2} + 132 T^{3} - 995 T^{4} + 132 p T^{5} + 7 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_2^2:C_4$ | \( 1 + 15 T + 69 T^{2} + 95 T^{3} + 36 T^{4} + 95 p T^{5} + 69 p^{2} T^{6} + 15 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_2^2:C_4$ | \( 1 + 6 T - 21 T^{2} - 248 T^{3} - 471 T^{4} - 248 p T^{5} - 21 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2^2:C_4$ | \( 1 + 6 T + 35 T^{2} + 84 T^{3} - 371 T^{4} + 84 p T^{5} + 35 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 - 6 T + 50 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $C_4\times C_2$ | \( 1 + 10 T + 13 T^{2} + 200 T^{3} + 3549 T^{4} + 200 p T^{5} + 13 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2^2:C_4$ | \( 1 + 14 T + 43 T^{2} - 650 T^{3} - 8799 T^{4} - 650 p T^{5} + 43 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_2^2:C_4$ | \( 1 + 6 T - 43 T^{2} - 102 T^{3} + 3025 T^{4} - 102 p T^{5} - 43 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_4\times C_2$ | \( 1 - p T^{2} + p^{2} T^{4} - p^{3} T^{6} + p^{4} T^{8} \) |
| 67 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 71 | $C_2^2:C_4$ | \( 1 + 24 T + 185 T^{2} + 456 T^{3} + 49 T^{4} + 456 p T^{5} + 185 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2^2:C_4$ | \( 1 + 22 T + 111 T^{2} - 1054 T^{3} - 17431 T^{4} - 1054 p T^{5} + 111 p^{2} T^{6} + 22 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_4\times C_2$ | \( 1 - 10 T + 21 T^{2} + 580 T^{3} - 7459 T^{4} + 580 p T^{5} + 21 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_2^2:C_4$ | \( 1 + 2 T + 121 T^{2} + 736 T^{3} + 7989 T^{4} + 736 p T^{5} + 121 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 - 9 T + 197 T^{2} - 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $C_4\times C_2$ | \( 1 - 6 T - 61 T^{2} + 948 T^{3} + 229 T^{4} + 948 p T^{5} - 61 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−9.054329496331560862426876153933, −8.595524802711426334149646827398, −8.258712057840916865504512551967, −8.044853697327831252247979152892, −7.73411596900232356257757697342, −7.38802888192933160720681712939, −7.23884287980394428942887592653, −6.71850506977507369840778928231, −6.64319819542584380170959766054, −6.58081179938490770254020466968, −6.10142010350435782309384236173, −5.48250980032149488080858725882, −5.35074586395700497433580769644, −5.20582693833899828085690554349, −5.07557373058760816558962433311, −4.40650497033866931382246944218, −4.38119761209234627735904029778, −4.05302203030036290309211996989, −3.90352337734420169249135108571, −3.25359258629615319118096112760, −3.14212594990844433394571175096, −3.02717827356592508391798936121, −2.82617140227775771838161795188, −1.50658874267227019928295598953, −1.38398853279145817269965663805,
1.38398853279145817269965663805, 1.50658874267227019928295598953, 2.82617140227775771838161795188, 3.02717827356592508391798936121, 3.14212594990844433394571175096, 3.25359258629615319118096112760, 3.90352337734420169249135108571, 4.05302203030036290309211996989, 4.38119761209234627735904029778, 4.40650497033866931382246944218, 5.07557373058760816558962433311, 5.20582693833899828085690554349, 5.35074586395700497433580769644, 5.48250980032149488080858725882, 6.10142010350435782309384236173, 6.58081179938490770254020466968, 6.64319819542584380170959766054, 6.71850506977507369840778928231, 7.23884287980394428942887592653, 7.38802888192933160720681712939, 7.73411596900232356257757697342, 8.044853697327831252247979152892, 8.258712057840916865504512551967, 8.595524802711426334149646827398, 9.054329496331560862426876153933
