| L(s) = 1 | − 2·3-s + 2·5-s + 2·7-s + 4·11-s + 10·13-s − 4·15-s + 4·17-s + 6·19-s − 4·21-s − 4·25-s + 2·27-s + 4·29-s − 8·31-s − 8·33-s + 4·35-s + 4·37-s − 20·39-s + 4·41-s + 4·43-s + 8·47-s + 3·49-s − 8·51-s + 24·53-s + 8·55-s − 12·57-s − 2·59-s + 2·61-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.894·5-s + 0.755·7-s + 1.20·11-s + 2.77·13-s − 1.03·15-s + 0.970·17-s + 1.37·19-s − 0.872·21-s − 4/5·25-s + 0.384·27-s + 0.742·29-s − 1.43·31-s − 1.39·33-s + 0.676·35-s + 0.657·37-s − 3.20·39-s + 0.624·41-s + 0.609·43-s + 1.16·47-s + 3/7·49-s − 1.12·51-s + 3.29·53-s + 1.07·55-s − 1.58·57-s − 0.260·59-s + 0.256·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3211264 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3211264 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.205221623\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.205221623\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 + 2 T + 4 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 5 | $D_{4}$ | \( 1 - 2 T + 8 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 4 T + 14 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 10 T + 48 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $D_{4}$ | \( 1 - 6 T + 44 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 4 T + 50 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $D_{4}$ | \( 1 - 4 T + 66 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 43 | $D_{4}$ | \( 1 - 4 T + 78 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 8 T + 62 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 59 | $D_{4}$ | \( 1 + 2 T + 44 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 2 T - 24 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 71 | $D_{4}$ | \( 1 + 8 T + 110 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 12 T + 134 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 8 T + 126 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 6 T + 100 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 4 T - 10 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 4 T + 150 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
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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
−9.325166230646684323680386837603, −9.082212968115080639358554541556, −8.758028166703045158032560102372, −8.495185751396153071299639606507, −7.79867310516911357133777532705, −7.45885972488868787048883042133, −7.12412055238422234496933349023, −6.44888816116357708296580399403, −6.00344741871775420859994475311, −5.85841686944782568620606814940, −5.61067644981718202860208955833, −5.38064094286548482556086175334, −4.46882833829330594826522999392, −4.16159996138097319539441839007, −3.59996945866311307851627166573, −3.30822418138074584916558933490, −2.42540600721553076586501107411, −1.71461318582810005087502364000, −1.10155717975019364145964280209, −0.955944972784133889138847899236,
0.955944972784133889138847899236, 1.10155717975019364145964280209, 1.71461318582810005087502364000, 2.42540600721553076586501107411, 3.30822418138074584916558933490, 3.59996945866311307851627166573, 4.16159996138097319539441839007, 4.46882833829330594826522999392, 5.38064094286548482556086175334, 5.61067644981718202860208955833, 5.85841686944782568620606814940, 6.00344741871775420859994475311, 6.44888816116357708296580399403, 7.12412055238422234496933349023, 7.45885972488868787048883042133, 7.79867310516911357133777532705, 8.495185751396153071299639606507, 8.758028166703045158032560102372, 9.082212968115080639358554541556, 9.325166230646684323680386837603
