| L(s) = 1 | − 2-s − 3·3-s − 3·5-s + 3·6-s − 6·7-s + 8-s + 2·9-s + 3·10-s − 6·11-s + 2·13-s + 6·14-s + 9·15-s − 16-s − 2·18-s − 3·19-s + 18·21-s + 6·22-s − 4·23-s − 3·24-s + 8·25-s − 2·26-s + 6·27-s − 5·29-s − 9·30-s − 3·31-s + 18·33-s + 18·35-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.73·3-s − 1.34·5-s + 1.22·6-s − 2.26·7-s + 0.353·8-s + 2/3·9-s + 0.948·10-s − 1.80·11-s + 0.554·13-s + 1.60·14-s + 2.32·15-s − 1/4·16-s − 0.471·18-s − 0.688·19-s + 3.92·21-s + 1.27·22-s − 0.834·23-s − 0.612·24-s + 8/5·25-s − 0.392·26-s + 1.15·27-s − 0.928·29-s − 1.64·30-s − 0.538·31-s + 3.13·33-s + 3.04·35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16180 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16180 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 2 | $C_2$ | \( 1 + T + T^{2} \) |
| 5 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 4 T + p T^{2} ) \) |
| 809 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 9 T + p T^{2} ) \) |
| good | 3 | $D_{4}$ | \( 1 + p T + 7 T^{2} + p^{2} T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 6 T + 20 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_4$ | \( 1 + 6 T + 26 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 2 T + 5 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 3 T + 22 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 4 T + 10 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 5 T + 15 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 3 T + 12 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 3 T - 21 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 2 T + 29 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 4 T + 83 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $D_{4}$ | \( 1 + 4 T + 49 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 15 T + 170 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 4 T + 55 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + T + 33 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 5 T + 122 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 4 T + 116 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 6 T - 16 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 17 T + 178 T^{2} - 17 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 21 T + 265 T^{2} + 21 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 9 T + 83 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−16.4458911553, −16.2635039524, −15.6617072502, −15.5270838099, −14.8676665167, −13.9352674861, −13.2169155297, −12.9818111043, −12.3667221200, −12.2324190219, −11.3101387011, −11.1001497207, −10.5268409930, −10.2358776259, −9.52355220871, −8.93485375163, −8.20480919678, −7.74679965938, −7.03000850278, −6.39717494630, −5.97102565360, −5.33712359852, −4.52359722854, −3.56724546562, −2.87813615328, 0, 0,
2.87813615328, 3.56724546562, 4.52359722854, 5.33712359852, 5.97102565360, 6.39717494630, 7.03000850278, 7.74679965938, 8.20480919678, 8.93485375163, 9.52355220871, 10.2358776259, 10.5268409930, 11.1001497207, 11.3101387011, 12.2324190219, 12.3667221200, 12.9818111043, 13.2169155297, 13.9352674861, 14.8676665167, 15.5270838099, 15.6617072502, 16.2635039524, 16.4458911553
