| L(s) = 1 | − 81·9-s + 312·11-s − 1.48e3·19-s − 5.14e3·29-s − 9.15e3·31-s − 2.04e4·41-s + 3.04e4·49-s − 9.73e4·59-s − 6.75e4·61-s + 7.65e4·71-s − 1.98e5·79-s + 6.56e3·81-s − 1.89e5·89-s − 2.52e4·99-s − 1.43e5·101-s − 8.38e4·109-s − 2.49e5·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 6.20e5·169-s + ⋯ |
| L(s) = 1 | − 1/3·9-s + 0.777·11-s − 0.940·19-s − 1.13·29-s − 1.71·31-s − 1.90·41-s + 1.81·49-s − 3.64·59-s − 2.32·61-s + 1.80·71-s − 3.57·79-s + 1/9·81-s − 2.53·89-s − 0.259·99-s − 1.39·101-s − 0.675·109-s − 1.54·121-s + 5.50e−6·127-s + 5.09e−6·131-s + 4.55e−6·137-s + 4.38e−6·139-s + 3.69e−6·149-s + 3.56e−6·151-s + 3.23e−6·157-s + 2.94e−6·163-s + 2.77e−6·167-s + 1.67·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 90000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 90000 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( 1 + p^{4} T^{2} \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( 1 - 622 p^{2} T^{2} + p^{10} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 156 T + p^{5} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 620086 T^{2} + p^{10} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 2221918 T^{2} + p^{10} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 740 T + p^{5} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 7227310 T^{2} + p^{10} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 2574 T + p^{5} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 4576 T + p^{5} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 10200890 T^{2} + p^{10} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 10230 T + p^{5} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 35321830 T^{2} + p^{10} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 457943518 T^{2} + p^{10} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 524586022 T^{2} + p^{10} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 48684 T + p^{5} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 33778 T + p^{5} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 2687831638 T^{2} + p^{10} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 38280 T + p^{5} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 2206265618 T^{2} + p^{10} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 99248 T + p^{5} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 7381504630 T^{2} + p^{10} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 94650 T + p^{5} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 17091469630 T^{2} + p^{10} T^{4} \) |
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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
−10.50368851208033655549351479642, −10.48447501838606394801843312713, −9.546164462414613901309659826995, −9.256245389933457551713324850720, −8.805620555463409931407131903313, −8.432484460047690455657370848601, −7.56188119221663308024632847513, −7.46863223642830756097818686575, −6.59116853558431347963630603672, −6.37215301929787067537687355052, −5.52992887160980915274315087837, −5.34600462011223155220723039630, −4.28367225842275723476253860414, −4.10224470976136495175682798209, −3.27964123098561208696478827857, −2.72520286852223662250709784831, −1.67388829984257524908138150959, −1.50224746074936812557620432983, 0, 0,
1.50224746074936812557620432983, 1.67388829984257524908138150959, 2.72520286852223662250709784831, 3.27964123098561208696478827857, 4.10224470976136495175682798209, 4.28367225842275723476253860414, 5.34600462011223155220723039630, 5.52992887160980915274315087837, 6.37215301929787067537687355052, 6.59116853558431347963630603672, 7.46863223642830756097818686575, 7.56188119221663308024632847513, 8.432484460047690455657370848601, 8.805620555463409931407131903313, 9.256245389933457551713324850720, 9.546164462414613901309659826995, 10.48447501838606394801843312713, 10.50368851208033655549351479642
