| L(s) = 1 | + 3·3-s − 2·7-s + 6·9-s − 3·11-s + 2·13-s − 6·17-s + 2·19-s − 6·21-s + 6·23-s + 5·25-s + 9·27-s + 6·29-s + 4·31-s − 9·33-s + 8·37-s + 6·39-s − 9·41-s − 43-s + 6·47-s + 7·49-s − 18·51-s − 24·53-s + 6·57-s + 3·59-s + 8·61-s − 12·63-s + 5·67-s + ⋯ |
| L(s) = 1 | + 1.73·3-s − 0.755·7-s + 2·9-s − 0.904·11-s + 0.554·13-s − 1.45·17-s + 0.458·19-s − 1.30·21-s + 1.25·23-s + 25-s + 1.73·27-s + 1.11·29-s + 0.718·31-s − 1.56·33-s + 1.31·37-s + 0.960·39-s − 1.40·41-s − 0.152·43-s + 0.875·47-s + 49-s − 2.52·51-s − 3.29·53-s + 0.794·57-s + 0.390·59-s + 1.02·61-s − 1.51·63-s + 0.610·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 331776 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 331776 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.190221062\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.190221062\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - p T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 2 T - 3 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 3 T - 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 6 T + 13 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 6 T + 7 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 9 T + 40 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + T - 42 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 6 T - 11 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 3 T - 50 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 8 T + 3 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 12 T + 61 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 19 T + p T^{2} ) \) |
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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
−10.72305383898545842725791060104, −10.53451323853155045925429306928, −9.883836769546784319216918455702, −9.533581995551175569466065716540, −9.178052544955769210267289398166, −8.654396715069845586623535213226, −8.459661689076367849449258784608, −7.960549983500854019345144533333, −7.47059502115549113841695363803, −6.97653707379361751422972252768, −6.48424228187937617705928151915, −6.23998793287684594489211087629, −5.16673678918970595668852023714, −4.79436426235201640162779404909, −4.27671346041948067499332489713, −3.51590222255242477545000646547, −3.04339839537422185967313347664, −2.70223626288530671589686135578, −2.05369672378010761092922360868, −0.963878146994308968468362047498,
0.963878146994308968468362047498, 2.05369672378010761092922360868, 2.70223626288530671589686135578, 3.04339839537422185967313347664, 3.51590222255242477545000646547, 4.27671346041948067499332489713, 4.79436426235201640162779404909, 5.16673678918970595668852023714, 6.23998793287684594489211087629, 6.48424228187937617705928151915, 6.97653707379361751422972252768, 7.47059502115549113841695363803, 7.960549983500854019345144533333, 8.459661689076367849449258784608, 8.654396715069845586623535213226, 9.178052544955769210267289398166, 9.533581995551175569466065716540, 9.883836769546784319216918455702, 10.53451323853155045925429306928, 10.72305383898545842725791060104
