| L(s) = 1 | − 2·2-s − 3-s + 2·4-s + 5-s + 2·6-s − 4·7-s − 4·8-s − 2·10-s − 6·11-s − 2·12-s − 6·13-s + 8·14-s − 15-s + 8·16-s − 6·17-s − 7·19-s + 2·20-s + 4·21-s + 12·22-s + 8·23-s + 4·24-s + 12·26-s + 27-s − 8·28-s − 7·29-s + 2·30-s + 18·31-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 0.577·3-s + 4-s + 0.447·5-s + 0.816·6-s − 1.51·7-s − 1.41·8-s − 0.632·10-s − 1.80·11-s − 0.577·12-s − 1.66·13-s + 2.13·14-s − 0.258·15-s + 2·16-s − 1.45·17-s − 1.60·19-s + 0.447·20-s + 0.872·21-s + 2.55·22-s + 1.66·23-s + 0.816·24-s + 2.35·26-s + 0.192·27-s − 1.51·28-s − 1.29·29-s + 0.365·30-s + 3.23·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 81225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T + T^{2} \) |
| 5 | $C_2$ | \( 1 - T + T^{2} \) |
| 19 | $C_2$ | \( 1 + 7 T + p T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 + p T + p T^{2} + p^{2} T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 6 T + 23 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 6 T + 19 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 8 T + 41 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 7 T + 20 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 6 T - 5 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 10 T + 57 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 4 T - 31 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 14 T + 143 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 3 T - 50 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 7 T - 12 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 4 T - 51 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 7 T - 22 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 2 T - 69 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 5 T - 54 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 3 T - 80 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 12 T + 47 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
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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
−11.44041867504039776305930414875, −10.98736794779768087415666667552, −10.32776115212624264008329999657, −10.13064529512781838969351215215, −9.754558477423477865376608101697, −9.423290489395488255228182447854, −8.601920224152026676854597448667, −8.544249894031179165270497702281, −7.86714648183370700147441033604, −7.11203936558606311489231782150, −6.63694526306519805057173474227, −6.38110975283015426625228134665, −5.78268758879400743688009078343, −4.85758787905193926360500455544, −4.74765612346183049747446150585, −3.02371113009014549865396421030, −2.96082758001508475377852847490, −2.08658067997107687672058176598, 0, 0,
2.08658067997107687672058176598, 2.96082758001508475377852847490, 3.02371113009014549865396421030, 4.74765612346183049747446150585, 4.85758787905193926360500455544, 5.78268758879400743688009078343, 6.38110975283015426625228134665, 6.63694526306519805057173474227, 7.11203936558606311489231782150, 7.86714648183370700147441033604, 8.544249894031179165270497702281, 8.601920224152026676854597448667, 9.423290489395488255228182447854, 9.754558477423477865376608101697, 10.13064529512781838969351215215, 10.32776115212624264008329999657, 10.98736794779768087415666667552, 11.44041867504039776305930414875
