L(s) = 1 | + 2-s + 2·3-s + 4-s − 2·5-s + 2·6-s + 3·7-s + 8-s − 2·10-s − 2·11-s + 2·12-s − 7·13-s + 3·14-s − 4·15-s + 16-s − 2·20-s + 6·21-s − 2·22-s + 11·23-s + 2·24-s + 2·25-s − 7·26-s − 2·27-s + 3·28-s − 7·29-s − 4·30-s − 5·31-s + 32-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1.15·3-s + 1/2·4-s − 0.894·5-s + 0.816·6-s + 1.13·7-s + 0.353·8-s − 0.632·10-s − 0.603·11-s + 0.577·12-s − 1.94·13-s + 0.801·14-s − 1.03·15-s + 1/4·16-s − 0.447·20-s + 1.30·21-s − 0.426·22-s + 2.29·23-s + 0.408·24-s + 2/5·25-s − 1.37·26-s − 0.384·27-s + 0.566·28-s − 1.29·29-s − 0.730·30-s − 0.898·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13040 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13040 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.899891421\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.899891421\) |
\(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 - T \) |
| 5 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 3 T + p T^{2} ) \) |
| 163 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - 2 T + p T^{2} ) \) |
good | 3 | $D_{4}$ | \( 1 - 2 T + 4 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $D_{4}$ | \( 1 + 2 T + 10 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 20 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 29 | $D_{4}$ | \( 1 + 7 T + 52 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $D_{4}$ | \( 1 + 2 T - 42 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 9 T + 40 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 47 | $D_{4}$ | \( 1 - 12 T + 112 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 80 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 130 T^{2} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 14 T + 168 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 9 T + 124 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 6 T + 142 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 3 T + 112 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
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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
−16.0946460780, −15.3911800154, −15.0642852419, −14.6832521395, −14.6369206317, −13.9419590494, −13.3827133670, −12.8866754774, −12.3635201923, −11.8191135856, −11.3791306062, −10.8301635982, −10.3420076727, −9.45170557785, −8.86265417002, −8.47556773247, −7.69090165174, −7.34553049435, −7.06360776842, −5.62679434132, −5.03497619192, −4.62084112066, −3.59700385765, −2.89457705307, −2.13784832033,
2.13784832033, 2.89457705307, 3.59700385765, 4.62084112066, 5.03497619192, 5.62679434132, 7.06360776842, 7.34553049435, 7.69090165174, 8.47556773247, 8.86265417002, 9.45170557785, 10.3420076727, 10.8301635982, 11.3791306062, 11.8191135856, 12.3635201923, 12.8866754774, 13.3827133670, 13.9419590494, 14.6369206317, 14.6832521395, 15.0642852419, 15.3911800154, 16.0946460780