| L(s) = 1 | − 4-s − 2·5-s − 7-s + 2·11-s − 4·13-s + 16-s − 4·19-s + 2·20-s + 8·23-s + 2·25-s + 28-s + 2·29-s + 6·31-s + 2·35-s + 4·37-s − 7·41-s − 12·43-s − 2·44-s + 12·47-s − 4·49-s + 4·52-s − 2·53-s − 4·55-s − 6·59-s − 14·61-s − 64-s + 8·65-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 0.894·5-s − 0.377·7-s + 0.603·11-s − 1.10·13-s + 1/4·16-s − 0.917·19-s + 0.447·20-s + 1.66·23-s + 2/5·25-s + 0.188·28-s + 0.371·29-s + 1.07·31-s + 0.338·35-s + 0.657·37-s − 1.09·41-s − 1.82·43-s − 0.301·44-s + 1.75·47-s − 4/7·49-s + 0.554·52-s − 0.274·53-s − 0.539·55-s − 0.781·59-s − 1.79·61-s − 1/8·64-s + 0.992·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4506262244\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4506262244\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 7 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 8 T + p T^{2} ) \) |
| good | 3 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 4 T + 24 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 4 T + 28 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 29 | $D_{4}$ | \( 1 - 2 T - 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 6 T + 30 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 4 T + 22 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 12 T + 82 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 12 T + 86 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $D_{4}$ | \( 1 + 6 T + 98 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 14 T + 110 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 6 T - 2 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 10 T + 78 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $D_{4}$ | \( 1 + 6 T - 42 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
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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
−19.7854274497, −19.2152665015, −18.7669222262, −18.3123319672, −17.2619315042, −17.0006815982, −16.6974122140, −15.5708209761, −15.3061921311, −14.7437398912, −14.0885637163, −13.3729684625, −12.7135096559, −12.1872820047, −11.6158893171, −10.8347498224, −10.0867871734, −9.37174600295, −8.68010542173, −7.98134724340, −7.07687294295, −6.42494862709, −5.03378457064, −4.32725550954, −3.10352497330,
3.10352497330, 4.32725550954, 5.03378457064, 6.42494862709, 7.07687294295, 7.98134724340, 8.68010542173, 9.37174600295, 10.0867871734, 10.8347498224, 11.6158893171, 12.1872820047, 12.7135096559, 13.3729684625, 14.0885637163, 14.7437398912, 15.3061921311, 15.5708209761, 16.6974122140, 17.0006815982, 17.2619315042, 18.3123319672, 18.7669222262, 19.2152665015, 19.7854274497
