| L(s) = 1 | − 2-s + 3-s + 4-s − 5-s − 6-s + 3·7-s − 3·8-s − 9-s + 10-s + 4·11-s + 12-s − 13-s − 3·14-s − 15-s + 16-s − 4·17-s + 18-s − 2·19-s − 20-s + 3·21-s − 4·22-s + 2·23-s − 3·24-s + 25-s + 26-s + 3·28-s − 29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s + 1.13·7-s − 1.06·8-s − 1/3·9-s + 0.316·10-s + 1.20·11-s + 0.288·12-s − 0.277·13-s − 0.801·14-s − 0.258·15-s + 1/4·16-s − 0.970·17-s + 0.235·18-s − 0.458·19-s − 0.223·20-s + 0.654·21-s − 0.852·22-s + 0.417·23-s − 0.612·24-s + 1/5·25-s + 0.196·26-s + 0.566·28-s − 0.185·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4069 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4069 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6749763051\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6749763051\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 13 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 2 T + p T^{2} ) \) |
| 313 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - 10 T + p T^{2} ) \) |
| good | 2 | $D_{4}$ | \( 1 + T + p T^{3} + p^{2} T^{4} \) |
| 3 | $D_{4}$ | \( 1 - T + 2 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + T + p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + p T^{2} ) \) |
| 11 | $C_4$ | \( 1 - 4 T + 6 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $D_{4}$ | \( 1 - 2 T + 14 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + T + 24 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 3 T - 18 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 3 T + 24 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 11 T + 80 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 47 | $D_{4}$ | \( 1 + 8 T + 30 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 10 T + 90 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 4 T + 70 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $D_{4}$ | \( 1 + 3 T + 34 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 8 T + 102 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 7 T + 132 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 8 T - 2 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 4 T + 142 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 97 | $C_2^2$ | \( 1 + 126 T^{2} + p^{2} T^{4} \) |
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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
−17.6933000359, −17.3997572945, −16.9811615228, −16.2448651421, −15.7298604797, −15.0191711441, −14.7841521620, −14.3845864490, −13.7598677796, −12.9574832886, −12.3279840962, −11.6374025849, −11.3023564079, −10.9440042934, −9.91897747528, −9.20178303158, −8.87518803156, −8.28177241515, −7.79644667291, −6.75767250449, −6.44136219210, −5.16735917822, −4.31100178652, −3.23116167279, −2.00202116460,
2.00202116460, 3.23116167279, 4.31100178652, 5.16735917822, 6.44136219210, 6.75767250449, 7.79644667291, 8.28177241515, 8.87518803156, 9.20178303158, 9.91897747528, 10.9440042934, 11.3023564079, 11.6374025849, 12.3279840962, 12.9574832886, 13.7598677796, 14.3845864490, 14.7841521620, 15.0191711441, 15.7298604797, 16.2448651421, 16.9811615228, 17.3997572945, 17.6933000359
