| L(s) = 1 | + 220·4-s − 729·9-s − 1.89e3·11-s + 3.20e4·16-s + 1.72e4·19-s − 7.30e4·29-s − 5.53e5·31-s − 1.60e5·36-s − 1.25e6·41-s − 4.17e5·44-s + 1.64e6·49-s − 2.61e6·59-s + 6.01e5·61-s + 3.43e6·64-s + 1.11e7·71-s + 3.79e6·76-s + 1.38e7·79-s + 5.31e5·81-s + 1.70e7·89-s + 1.38e6·99-s + 2.39e7·101-s − 8.04e6·109-s − 1.60e7·116-s − 3.62e7·121-s − 1.21e8·124-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | + 1.71·4-s − 1/3·9-s − 0.429·11-s + 1.95·16-s + 0.576·19-s − 0.555·29-s − 3.33·31-s − 0.572·36-s − 2.85·41-s − 0.738·44-s + 1.99·49-s − 1.65·59-s + 0.339·61-s + 1.63·64-s + 3.68·71-s + 0.991·76-s + 3.15·79-s + 1/9·81-s + 2.56·89-s + 0.143·99-s + 2.31·101-s − 0.594·109-s − 0.955·116-s − 1.86·121-s − 5.73·124-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s+7/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(3.481659892\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.481659892\) |
| \(L(\frac{9}{2})\) |
|
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 | $C_2$ | \( 1 + p^{6} T^{2} \) |
| 5 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 - 55 p^{2} T^{2} + p^{14} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 1642990 T^{2} + p^{14} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 948 T + p^{7} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 99507430 T^{2} + p^{14} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 14912350 T^{2} + p^{14} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 8620 T + p^{7} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 6575927950 T^{2} + p^{14} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 36510 T + p^{7} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 276808 T + p^{7} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 117757541590 T^{2} + p^{14} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 629718 T + p^{7} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 73353986230 T^{2} + p^{14} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 673012017310 T^{2} + p^{14} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 2166188628310 T^{2} + p^{14} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 1306380 T + p^{7} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 300662 T + p^{7} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 11864126735110 T^{2} + p^{14} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 5560632 T + p^{7} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 20220411515470 T^{2} + p^{14} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 6913720 T + p^{7} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 35116178923750 T^{2} + p^{14} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 8528310 T + p^{7} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 83683923565630 T^{2} + p^{14} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.83730537304705295522036022454, −12.49498826873465102977703193203, −12.35637536140373273048413582767, −11.58635786181038026526055029379, −11.21710719538461534120833070799, −10.64259545087159780556239378648, −10.34620780551102606573043851005, −9.368430594908591086876722559669, −8.916916967159557663492250418692, −7.76473550139272299638928441200, −7.70540520776019758486919745325, −6.83232463280622585182263077114, −6.45278599336597204987675266092, −5.40808952445470138340703521596, −5.28058417901085053691375727390, −3.62202687984716412447962697966, −3.30969596766132656198520117453, −2.10109444820072015065824644526, −1.89559466240585282853950335937, −0.59492147725570881658289728348,
0.59492147725570881658289728348, 1.89559466240585282853950335937, 2.10109444820072015065824644526, 3.30969596766132656198520117453, 3.62202687984716412447962697966, 5.28058417901085053691375727390, 5.40808952445470138340703521596, 6.45278599336597204987675266092, 6.83232463280622585182263077114, 7.70540520776019758486919745325, 7.76473550139272299638928441200, 8.916916967159557663492250418692, 9.368430594908591086876722559669, 10.34620780551102606573043851005, 10.64259545087159780556239378648, 11.21710719538461534120833070799, 11.58635786181038026526055029379, 12.35637536140373273048413582767, 12.49498826873465102977703193203, 13.83730537304705295522036022454
