| L(s) = 1 | + 2-s + 2·4-s + 8·5-s + 7-s + 5·8-s + 8·10-s − 4·11-s − 13-s + 14-s + 5·16-s − 6·17-s − 4·19-s + 16·20-s − 4·22-s − 12·23-s + 38·25-s − 26-s + 2·28-s + 2·29-s − 3·31-s + 10·32-s − 6·34-s + 8·35-s − 3·37-s − 4·38-s + 40·40-s − 2·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 4-s + 3.57·5-s + 0.377·7-s + 1.76·8-s + 2.52·10-s − 1.20·11-s − 0.277·13-s + 0.267·14-s + 5/4·16-s − 1.45·17-s − 0.917·19-s + 3.57·20-s − 0.852·22-s − 2.50·23-s + 38/5·25-s − 0.196·26-s + 0.377·28-s + 0.371·29-s − 0.538·31-s + 1.76·32-s − 1.02·34-s + 1.35·35-s − 0.493·37-s − 0.648·38-s + 6.32·40-s − 0.312·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 321489 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 321489 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.247379655\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.247379655\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_2$ | \( 1 - T + p T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - T - T^{2} - p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + T - 12 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 6 T + 19 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 4 T - 3 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 2 T - 25 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 3 T - 22 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 3 T - 28 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 2 T - 37 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - T - 42 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 6 T - 11 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 6 T - 17 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 6 T - 23 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 5 T - 36 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 7 T - 18 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 6 T - 37 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 11 T + 42 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 6 T - 47 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 4 T - 73 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 9 T - 16 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
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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
−10.56575698242672125480583586881, −10.49061482550160732493593987685, −10.24985323991674567022068225906, −9.976986486308580069177072648560, −9.201919348009514589724953046630, −9.056032715321672295082332370771, −8.138493897444691604258050934501, −8.001301790541918977629598814131, −6.92113322002411786060714360600, −6.89457929772165343952836471243, −6.11171638363288355979697029106, −6.04149331496664660218454049196, −5.52355297196891744851029340591, −4.89126320619457621862929518741, −4.76400825215138679525364290619, −3.91261863832609813463671141856, −2.57193976950475421936774777077, −2.46872887508172216304132251587, −1.82812592323858053351951550137, −1.71528085728846256494530826959,
1.71528085728846256494530826959, 1.82812592323858053351951550137, 2.46872887508172216304132251587, 2.57193976950475421936774777077, 3.91261863832609813463671141856, 4.76400825215138679525364290619, 4.89126320619457621862929518741, 5.52355297196891744851029340591, 6.04149331496664660218454049196, 6.11171638363288355979697029106, 6.89457929772165343952836471243, 6.92113322002411786060714360600, 8.001301790541918977629598814131, 8.138493897444691604258050934501, 9.056032715321672295082332370771, 9.201919348009514589724953046630, 9.976986486308580069177072648560, 10.24985323991674567022068225906, 10.49061482550160732493593987685, 10.56575698242672125480583586881
