| L(s) = 1 | − 2-s − 3-s − 4-s + 5-s + 6-s − 2·7-s + 8-s − 10-s + 4·11-s + 12-s + 7·13-s + 2·14-s − 15-s − 16-s + 17-s + 4·19-s − 20-s + 2·21-s − 4·22-s + 5·23-s − 24-s + 25-s − 7·26-s − 2·27-s + 2·28-s − 7·29-s + 30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s − 1/2·4-s + 0.447·5-s + 0.408·6-s − 0.755·7-s + 0.353·8-s − 0.316·10-s + 1.20·11-s + 0.288·12-s + 1.94·13-s + 0.534·14-s − 0.258·15-s − 1/4·16-s + 0.242·17-s + 0.917·19-s − 0.223·20-s + 0.436·21-s − 0.852·22-s + 1.04·23-s − 0.204·24-s + 1/5·25-s − 1.37·26-s − 0.384·27-s + 0.377·28-s − 1.29·29-s + 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6827 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6827 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5038328717\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5038328717\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 6827 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 147 T + p T^{2} ) \) |
| good | 2 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 3 | $D_{4}$ | \( 1 + T + T^{2} + p T^{3} + p^{2} T^{4} \) |
| 5 | $D_{4}$ | \( 1 - T - p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 2 T + 4 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 13 | $D_{4}$ | \( 1 - 7 T + 28 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - T + 18 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 5 T + 15 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 7 T + 38 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 4 T + 16 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 10 T + 70 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_4$ | \( 1 - 9 T + 71 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 3 T - p T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $D_{4}$ | \( 1 + 10 T + 110 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 98 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 67 | $D_{4}$ | \( 1 - 6 T + 38 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 3 T - 83 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 9 T + 96 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 7 T + 33 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 8 T + 112 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 3 T - 32 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 3 T + 116 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
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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
−17.0910378974, −16.7946024123, −16.2276029742, −15.6734925563, −15.3536437509, −14.3626846345, −14.0145570535, −13.4961604539, −13.0473539630, −12.4703268543, −11.7751233863, −11.1743744345, −10.8923944745, −10.0173106200, −9.43935710905, −9.11774610246, −8.71659595963, −7.86160965675, −6.98744155608, −6.29197058772, −5.92957940621, −5.04146826213, −3.96139132455, −3.25944489238, −1.26452726685,
1.26452726685, 3.25944489238, 3.96139132455, 5.04146826213, 5.92957940621, 6.29197058772, 6.98744155608, 7.86160965675, 8.71659595963, 9.11774610246, 9.43935710905, 10.0173106200, 10.8923944745, 11.1743744345, 11.7751233863, 12.4703268543, 13.0473539630, 13.4961604539, 14.0145570535, 14.3626846345, 15.3536437509, 15.6734925563, 16.2276029742, 16.7946024123, 17.0910378974
