| L(s) = 1 | − 2·5-s − 2·7-s − 3·9-s + 2·11-s − 17-s − 3·19-s + 3·23-s − 25-s − 4·29-s − 3·31-s + 4·35-s + 6·37-s + 12·41-s − 6·43-s + 6·45-s + 4·47-s − 2·49-s + 3·53-s − 4·55-s − 6·59-s + 3·61-s + 6·63-s + 8·67-s + 2·71-s + 12·73-s − 4·77-s + 3·79-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 0.755·7-s − 9-s + 0.603·11-s − 0.242·17-s − 0.688·19-s + 0.625·23-s − 1/5·25-s − 0.742·29-s − 0.538·31-s + 0.676·35-s + 0.986·37-s + 1.87·41-s − 0.914·43-s + 0.894·45-s + 0.583·47-s − 2/7·49-s + 0.412·53-s − 0.539·55-s − 0.781·59-s + 0.384·61-s + 0.755·63-s + 0.977·67-s + 0.237·71-s + 1.40·73-s − 0.455·77-s + 0.337·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 172800 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 172800 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8494900182\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8494900182\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + p T^{2} \) |
| 5 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| good | 7 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $D_{4}$ | \( 1 - 2 T + 6 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $D_{4}$ | \( 1 + T + 12 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 3 T + 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 3 T + 10 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $D_{4}$ | \( 1 - 6 T + 42 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 47 | $D_{4}$ | \( 1 - 4 T - 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 3 T + 64 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 6 T + 110 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 3 T + 96 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 8 T + 110 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 2 T - 58 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 12 T + 102 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 3 T - 18 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 89 | $C_2^2$ | \( 1 + 110 T^{2} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 6 T - 14 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
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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
−13.4640745548, −13.1037564337, −12.5405863092, −12.4475754994, −11.7675007441, −11.2992148896, −11.1606020319, −10.7347765523, −10.0367915013, −9.46341042648, −9.24772969438, −8.72094667844, −8.23311947547, −7.79257472207, −7.30058369481, −6.69472483660, −6.32781139254, −5.74410157121, −5.27623369871, −4.39549042364, −4.01348268823, −3.41274403561, −2.82273666976, −2.02697384116, −0.587552058637,
0.587552058637, 2.02697384116, 2.82273666976, 3.41274403561, 4.01348268823, 4.39549042364, 5.27623369871, 5.74410157121, 6.32781139254, 6.69472483660, 7.30058369481, 7.79257472207, 8.23311947547, 8.72094667844, 9.24772969438, 9.46341042648, 10.0367915013, 10.7347765523, 11.1606020319, 11.2992148896, 11.7675007441, 12.4475754994, 12.5405863092, 13.1037564337, 13.4640745548
