| L(s) = 1 | + 3·3-s + 4·9-s + 11-s + 4·17-s + 4·19-s − 4·25-s + 3·33-s − 8·41-s − 2·43-s + 2·49-s + 12·51-s + 12·57-s + 8·59-s + 4·67-s + 4·73-s − 12·75-s − 11·81-s + 16·89-s − 20·97-s + 4·99-s + 2·107-s − 28·113-s − 8·121-s − 24·123-s + 127-s − 6·129-s + 131-s + ⋯ |
| L(s) = 1 | + 1.73·3-s + 4/3·9-s + 0.301·11-s + 0.970·17-s + 0.917·19-s − 4/5·25-s + 0.522·33-s − 1.24·41-s − 0.304·43-s + 2/7·49-s + 1.68·51-s + 1.58·57-s + 1.04·59-s + 0.488·67-s + 0.468·73-s − 1.38·75-s − 1.22·81-s + 1.69·89-s − 2.03·97-s + 0.402·99-s + 0.193·107-s − 2.63·113-s − 0.727·121-s − 2.16·123-s + 0.0887·127-s − 0.528·129-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 67584 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 67584 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.622785460\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.622785460\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - 2 T + p T^{2} ) \) |
| 11 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 2 T + p T^{2} ) \) |
| good | 5 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 + 20 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 28 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 - 24 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 54 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 + 8 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
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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
−9.736057125711002315625144741818, −9.329770215233763856612161487845, −8.862485394163136155242063527204, −8.310642620167344857634327566288, −7.936271900123213334998403179767, −7.54566878648857010646494731987, −6.94321724319471127237538445045, −6.33388347191979292565085545900, −5.49088124384102158234193946447, −5.10277558756072013950614946800, −4.01916199198331923073217822530, −3.65202188248691397587073321193, −3.03962451446509644105467547253, −2.34874852387974852840455771358, −1.43025127183604391862407836281,
1.43025127183604391862407836281, 2.34874852387974852840455771358, 3.03962451446509644105467547253, 3.65202188248691397587073321193, 4.01916199198331923073217822530, 5.10277558756072013950614946800, 5.49088124384102158234193946447, 6.33388347191979292565085545900, 6.94321724319471127237538445045, 7.54566878648857010646494731987, 7.936271900123213334998403179767, 8.310642620167344857634327566288, 8.862485394163136155242063527204, 9.329770215233763856612161487845, 9.736057125711002315625144741818
