| L(s) = 1 | − 2·3-s − 4-s − 2·5-s − 7-s + 2·8-s + 2·9-s + 2·12-s + 4·13-s + 4·15-s + 16-s − 4·17-s − 2·19-s + 2·20-s + 2·21-s − 4·23-s − 4·24-s + 2·25-s − 6·27-s + 28-s + 6·29-s + 4·31-s − 4·32-s + 2·35-s − 2·36-s + 4·37-s − 8·39-s − 4·40-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 1/2·4-s − 0.894·5-s − 0.377·7-s + 0.707·8-s + 2/3·9-s + 0.577·12-s + 1.10·13-s + 1.03·15-s + 1/4·16-s − 0.970·17-s − 0.458·19-s + 0.447·20-s + 0.436·21-s − 0.834·23-s − 0.816·24-s + 2/5·25-s − 1.15·27-s + 0.188·28-s + 1.11·29-s + 0.718·31-s − 0.707·32-s + 0.338·35-s − 1/3·36-s + 0.657·37-s − 1.28·39-s − 0.632·40-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 574 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 574 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2886587543\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2886587543\) |
| \(L(\frac{3}{2})\) |
|
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$ | \( ( 1 - T )( 1 + T + p T^{2} ) \) |
| 7 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 2 T + p T^{2} ) \) |
| good | 3 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 4 T + 6 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $D_{4}$ | \( 1 + 2 T + 18 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) |
| 31 | $D_{4}$ | \( 1 - 4 T + 6 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 4 T - 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 4 T + 6 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $D_{4}$ | \( 1 + 2 T - 38 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 6 T - 2 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 6 T - 14 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $D_{4}$ | \( 1 - 8 T + 78 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 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
−19.6733132147, −19.2894810758, −18.7091307297, −17.8708865577, −17.6960182246, −16.8518514821, −16.2735421554, −15.8505407017, −15.3180145253, −14.3426984432, −13.5290037260, −13.0723731511, −12.3156957452, −11.5725833897, −11.0601290334, −10.4541609560, −9.56623783527, −8.50715996695, −7.85008095901, −6.70727794107, −5.99154001731, −4.71800513845, −3.93159440622,
3.93159440622, 4.71800513845, 5.99154001731, 6.70727794107, 7.85008095901, 8.50715996695, 9.56623783527, 10.4541609560, 11.0601290334, 11.5725833897, 12.3156957452, 13.0723731511, 13.5290037260, 14.3426984432, 15.3180145253, 15.8505407017, 16.2735421554, 16.8518514821, 17.6960182246, 17.8708865577, 18.7091307297, 19.2894810758, 19.6733132147
