| L(s) = 1 | + 4·4-s + 5·9-s − 6·11-s + 12·16-s + 4·19-s − 6·29-s + 8·31-s + 20·36-s + 24·41-s − 24·44-s − 16·61-s + 32·64-s + 16·76-s + 2·79-s + 16·81-s − 24·89-s − 30·99-s − 12·101-s + 14·109-s − 24·116-s + 5·121-s + 32·124-s + 127-s + 131-s + 137-s + 139-s + 60·144-s + ⋯ |
| L(s) = 1 | + 2·4-s + 5/3·9-s − 1.80·11-s + 3·16-s + 0.917·19-s − 1.11·29-s + 1.43·31-s + 10/3·36-s + 3.74·41-s − 3.61·44-s − 2.04·61-s + 4·64-s + 1.83·76-s + 0.225·79-s + 16/9·81-s − 2.54·89-s − 3.01·99-s − 1.19·101-s + 1.34·109-s − 2.22·116-s + 5/11·121-s + 2.87·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 5·144-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1500625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1500625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.573808116\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.573808116\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 2 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 3 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 13 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 142 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 193 T^{2} + p^{2} T^{4} \) |
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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
−9.809012076975704178050638223042, −9.722895826047067835675412783330, −9.443343083244125355106283534189, −8.492023814308683359530528321013, −8.066021885086306340485902760396, −7.65212341123739983301926105703, −7.36200319427344189562770520535, −7.32848234581028406475988253106, −6.69932945099849126946627826212, −6.20114581194712439796224933213, −5.67177085070311137272905068414, −5.61613901951469594840923339883, −4.74508080399287726146407319234, −4.38994178042530406142810336985, −3.71546315800725028546040145024, −3.05243238467350390075692737071, −2.64881610092155115619941036006, −2.26000522287000529379967910861, −1.52293871530131897144994127778, −0.958770528599014897711384588392,
0.958770528599014897711384588392, 1.52293871530131897144994127778, 2.26000522287000529379967910861, 2.64881610092155115619941036006, 3.05243238467350390075692737071, 3.71546315800725028546040145024, 4.38994178042530406142810336985, 4.74508080399287726146407319234, 5.61613901951469594840923339883, 5.67177085070311137272905068414, 6.20114581194712439796224933213, 6.69932945099849126946627826212, 7.32848234581028406475988253106, 7.36200319427344189562770520535, 7.65212341123739983301926105703, 8.066021885086306340485902760396, 8.492023814308683359530528321013, 9.443343083244125355106283534189, 9.722895826047067835675412783330, 9.809012076975704178050638223042
