| L(s) = 1 | − 2·3-s − 7-s + 9-s − 4·13-s + 2·17-s + 2·19-s + 2·21-s + 5·23-s − 5·25-s + 4·27-s − 5·29-s − 2·31-s + 6·37-s + 8·39-s − 41-s + 12·43-s − 47-s + 49-s − 4·51-s − 4·53-s − 4·57-s + 4·59-s − 3·61-s − 63-s − 15·67-s − 10·69-s − 8·71-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 0.377·7-s + 1/3·9-s − 1.10·13-s + 0.485·17-s + 0.458·19-s + 0.436·21-s + 1.04·23-s − 25-s + 0.769·27-s − 0.928·29-s − 0.359·31-s + 0.986·37-s + 1.28·39-s − 0.156·41-s + 1.82·43-s − 0.145·47-s + 1/7·49-s − 0.560·51-s − 0.549·53-s − 0.529·57-s + 0.520·59-s − 0.384·61-s − 0.125·63-s − 1.83·67-s − 1.20·69-s − 0.949·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 129600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 129600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | | \( 1 \) |
| 3 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 5 | $C_2$ | \( 1 + p T^{2} \) |
| good | 7 | $D_{4}$ | \( 1 + T + p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 4 T + 24 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 2 T + 32 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 2 T + 22 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 5 T + 28 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 5 T + 28 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 2 T + 34 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 6 T + 28 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + T + 56 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 12 T + 82 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + T + 20 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 4 T + 70 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 4 T + 34 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 67 | $D_{4}$ | \( 1 + 15 T + 132 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 8 T + 106 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 2 T + 42 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 6 T + 126 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + T + 60 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - T - 20 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 74 T^{2} + p^{2} T^{4} \) |
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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
−14.0985065859, −13.4408625657, −13.0772554860, −12.6920112553, −12.1503355897, −11.9078301904, −11.5177869749, −10.9193895260, −10.6939600517, −10.1425523489, −9.55680706912, −9.33158724017, −8.83950239563, −7.91258136164, −7.65606197488, −7.13011248732, −6.63982274375, −5.96055441335, −5.63744043130, −5.18452378859, −4.54224995682, −3.96242549999, −3.07108642950, −2.48848214398, −1.26701386764, 0,
1.26701386764, 2.48848214398, 3.07108642950, 3.96242549999, 4.54224995682, 5.18452378859, 5.63744043130, 5.96055441335, 6.63982274375, 7.13011248732, 7.65606197488, 7.91258136164, 8.83950239563, 9.33158724017, 9.55680706912, 10.1425523489, 10.6939600517, 10.9193895260, 11.5177869749, 11.9078301904, 12.1503355897, 12.6920112553, 13.0772554860, 13.4408625657, 14.0985065859
