| L(s) = 1 | + 2-s + 2·3-s − 4-s + 2·5-s + 2·6-s − 7-s − 8-s + 2·9-s + 2·10-s + 3·11-s − 2·12-s − 2·13-s − 14-s + 4·15-s − 16-s + 17-s + 2·18-s + 13·19-s − 2·20-s − 2·21-s + 3·22-s − 5·23-s − 2·24-s + 2·25-s − 2·26-s + 6·27-s + 28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.15·3-s − 1/2·4-s + 0.894·5-s + 0.816·6-s − 0.377·7-s − 0.353·8-s + 2/3·9-s + 0.632·10-s + 0.904·11-s − 0.577·12-s − 0.554·13-s − 0.267·14-s + 1.03·15-s − 1/4·16-s + 0.242·17-s + 0.471·18-s + 2.98·19-s − 0.447·20-s − 0.436·21-s + 0.639·22-s − 1.04·23-s − 0.408·24-s + 2/5·25-s − 0.392·26-s + 1.15·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 170255 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 170255 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.736055780\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.736055780\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 3 T + p T^{2} ) \) |
| 17 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + p T^{2} ) \) |
| 2003 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 32 T + p T^{2} ) \) |
| good | 2 | $C_2$$\times$$C_2$ | \( ( 1 - p T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 3 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + T + 6 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 3 T + 10 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 5 T + p T^{2} ) \) |
| 23 | $D_{4}$ | \( 1 + 5 T + 38 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 9 T + 68 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 4 T + 14 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 11 T + 100 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 3 T + 36 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 2 T - 10 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 9 T + 38 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 59 | $D_{4}$ | \( 1 + 13 T + 86 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 4 T + 78 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 6 T + 14 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 9 T + 130 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 4 T + 60 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 14 T + 180 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 6 T + 14 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 4 T + 150 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + T - 32 T^{2} + p T^{3} + p^{2} T^{4} \) |
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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
−13.7660931265, −13.4229490731, −12.8800494541, −12.2919619107, −12.0511811027, −11.7556216730, −10.9233514576, −10.3536576266, −9.80681860399, −9.70826077877, −9.19079871861, −8.89668464451, −8.31755216243, −7.77126887623, −7.27346024310, −6.71696019387, −6.31150845985, −5.42646382425, −5.19156162999, −4.64926472708, −3.89050678343, −3.29785862266, −3.02060580579, −2.06593341777, −1.23282473562,
1.23282473562, 2.06593341777, 3.02060580579, 3.29785862266, 3.89050678343, 4.64926472708, 5.19156162999, 5.42646382425, 6.31150845985, 6.71696019387, 7.27346024310, 7.77126887623, 8.31755216243, 8.89668464451, 9.19079871861, 9.70826077877, 9.80681860399, 10.3536576266, 10.9233514576, 11.7556216730, 12.0511811027, 12.2919619107, 12.8800494541, 13.4229490731, 13.7660931265
