| L(s) = 1 | − 2·2-s − 5-s − 4·7-s + 4·8-s + 2·10-s − 3·11-s − 3·13-s + 8·14-s − 4·16-s − 3·19-s + 6·22-s − 7·23-s + 3·25-s + 6·26-s + 7·31-s + 4·35-s − 4·37-s + 6·38-s − 4·40-s + 5·41-s − 2·43-s + 14·46-s − 5·47-s + 8·49-s − 6·50-s + 7·53-s + 3·55-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 0.447·5-s − 1.51·7-s + 1.41·8-s + 0.632·10-s − 0.904·11-s − 0.832·13-s + 2.13·14-s − 16-s − 0.688·19-s + 1.27·22-s − 1.45·23-s + 3/5·25-s + 1.17·26-s + 1.25·31-s + 0.676·35-s − 0.657·37-s + 0.973·38-s − 0.632·40-s + 0.780·41-s − 0.304·43-s + 2.06·46-s − 0.729·47-s + 8/7·49-s − 0.848·50-s + 0.961·53-s + 0.404·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3483 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3483 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 43 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + T + p T^{2} ) \) |
| good | 2 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 5 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 3 T + 7 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 3 T + p T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 23 | $D_{4}$ | \( 1 + 7 T + 37 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 32 T^{2} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 7 T + 27 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 5 T + 28 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 5 T + 85 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $D_{4}$ | \( 1 + 3 T + 58 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_4$ | \( 1 - 4 T - 54 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $D_{4}$ | \( 1 - 8 T + 64 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $D_{4}$ | \( 1 + 21 T + 226 T^{2} + 21 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 5 T + 58 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−18.4236248895, −17.8532115246, −17.2295125219, −16.9516180136, −16.2409943771, −15.9272568332, −15.4195988865, −14.6714323218, −13.9795893337, −13.3923541115, −12.9125695911, −12.3686238385, −11.7981428403, −10.7928967397, −10.1562335525, −9.96331549114, −9.40351100870, −8.62970827931, −8.21449826043, −7.53858062812, −6.80418302966, −5.95986541631, −4.85597494509, −3.97680754253, −2.71256941905, 0,
2.71256941905, 3.97680754253, 4.85597494509, 5.95986541631, 6.80418302966, 7.53858062812, 8.21449826043, 8.62970827931, 9.40351100870, 9.96331549114, 10.1562335525, 10.7928967397, 11.7981428403, 12.3686238385, 12.9125695911, 13.3923541115, 13.9795893337, 14.6714323218, 15.4195988865, 15.9272568332, 16.2409943771, 16.9516180136, 17.2295125219, 17.8532115246, 18.4236248895
