| L(s) = 1 | − 2·2-s − 4·3-s + 2·4-s − 3·5-s + 8·6-s − 2·7-s + 7·9-s + 6·10-s − 6·11-s − 8·12-s − 4·13-s + 4·14-s + 12·15-s − 4·16-s − 8·17-s − 14·18-s + 3·19-s − 6·20-s + 8·21-s + 12·22-s + 23-s + 8·26-s − 4·27-s − 4·28-s − 5·29-s − 24·30-s + 31-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 2.30·3-s + 4-s − 1.34·5-s + 3.26·6-s − 0.755·7-s + 7/3·9-s + 1.89·10-s − 1.80·11-s − 2.30·12-s − 1.10·13-s + 1.06·14-s + 3.09·15-s − 16-s − 1.94·17-s − 3.29·18-s + 0.688·19-s − 1.34·20-s + 1.74·21-s + 2.55·22-s + 0.208·23-s + 1.56·26-s − 0.769·27-s − 0.755·28-s − 0.928·29-s − 4.38·30-s + 0.179·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10996 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10996 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + p T + p T^{2} \) |
| 2749 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - 30 T + p T^{2} ) \) |
| good | 3 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 5 | $D_{4}$ | \( 1 + 3 T + 9 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 2 T + 3 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 17 | $D_{4}$ | \( 1 + 8 T + 45 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 3 T + 3 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - T - 20 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 5 T + 25 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - T - 15 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 7 T + 41 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 4 T + 46 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - T - 2 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 5 T + 37 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 3 T - 20 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 8 T + 42 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 16 T + 165 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 10 T + 79 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 2 T + 66 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 7 T + 13 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 21 T + 223 T^{2} - 21 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 16 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 126 T^{2} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 23 T + 299 T^{2} + 23 p T^{3} + p^{2} T^{4} \) |
| 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
−17.0621504161, −16.4492648213, −16.2153182139, −15.8622267290, −15.3121782143, −15.1521499072, −13.7208121063, −13.3532921414, −12.6717090073, −12.1606757062, −11.7504171489, −11.2613295641, −10.8588389916, −10.5231389349, −9.96586894267, −9.32148118513, −8.57645049017, −7.89382919940, −7.30826879395, −6.95843081965, −6.24117647264, −5.29164669274, −4.99624086113, −4.05884747764, −2.54181939631, 0, 0,
2.54181939631, 4.05884747764, 4.99624086113, 5.29164669274, 6.24117647264, 6.95843081965, 7.30826879395, 7.89382919940, 8.57645049017, 9.32148118513, 9.96586894267, 10.5231389349, 10.8588389916, 11.2613295641, 11.7504171489, 12.1606757062, 12.6717090073, 13.3532921414, 13.7208121063, 15.1521499072, 15.3121782143, 15.8622267290, 16.2153182139, 16.4492648213, 17.0621504161
