L(s) = 1 | − 2·3-s − 4-s + 3·5-s − 4·7-s + 2·8-s + 3·9-s − 5·11-s + 2·12-s − 8·13-s − 6·15-s + 16-s + 4·17-s + 8·19-s − 3·20-s + 8·21-s + 4·23-s − 4·24-s + 2·25-s − 4·27-s + 4·28-s + 4·29-s − 4·32-s + 10·33-s − 12·35-s − 3·36-s − 4·37-s + 16·39-s + ⋯ |
L(s) = 1 | − 1.15·3-s − 1/2·4-s + 1.34·5-s − 1.51·7-s + 0.707·8-s + 9-s − 1.50·11-s + 0.577·12-s − 2.21·13-s − 1.54·15-s + 1/4·16-s + 0.970·17-s + 1.83·19-s − 0.670·20-s + 1.74·21-s + 0.834·23-s − 0.816·24-s + 2/5·25-s − 0.769·27-s + 0.755·28-s + 0.742·29-s − 0.707·32-s + 1.74·33-s − 2.02·35-s − 1/2·36-s − 0.657·37-s + 2.56·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 990 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 990 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.3859335527\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3859335527\) |
\(L(\frac{3}{2})\) |
|
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$$\times$$C_2$ | \( ( 1 - T )( 1 + T + p T^{2} ) \) |
| 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - 2 T + p T^{2} ) \) |
| 11 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 4 T + p T^{2} ) \) |
good | 7 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + 8 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−19.9164486365, −19.1104624590, −19.0798046347, −18.1058537571, −17.7512408724, −17.2359534775, −16.7727116030, −16.0129129656, −15.9590260302, −14.7742712184, −14.0069258505, −13.5527745379, −12.7606431103, −12.6461787614, −11.8560024471, −10.6789224512, −10.1733530745, −9.62624986142, −9.52345167581, −7.66488013442, −7.24765397307, −6.15685276164, −5.23920392625, −5.03220048416, −2.90847219286,
2.90847219286, 5.03220048416, 5.23920392625, 6.15685276164, 7.24765397307, 7.66488013442, 9.52345167581, 9.62624986142, 10.1733530745, 10.6789224512, 11.8560024471, 12.6461787614, 12.7606431103, 13.5527745379, 14.0069258505, 14.7742712184, 15.9590260302, 16.0129129656, 16.7727116030, 17.2359534775, 17.7512408724, 18.1058537571, 19.0798046347, 19.1104624590, 19.9164486365