| L(s) = 1 | + 3-s − 5-s + 7-s − 9-s + 3·11-s − 2·13-s − 15-s + 17-s − 3·19-s + 21-s − 7·23-s − 2·25-s − 11·29-s + 3·33-s − 35-s − 9·37-s − 2·39-s − 9·41-s + 11·43-s + 45-s + 4·47-s − 5·49-s + 51-s − 3·55-s − 3·57-s − 3·59-s − 7·61-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s + 0.377·7-s − 1/3·9-s + 0.904·11-s − 0.554·13-s − 0.258·15-s + 0.242·17-s − 0.688·19-s + 0.218·21-s − 1.45·23-s − 2/5·25-s − 2.04·29-s + 0.522·33-s − 0.169·35-s − 1.47·37-s − 0.320·39-s − 1.40·41-s + 1.67·43-s + 0.149·45-s + 0.583·47-s − 5/7·49-s + 0.140·51-s − 0.404·55-s − 0.397·57-s − 0.390·59-s − 0.896·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 216320 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216320 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 5 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| good | 3 | $D_{4}$ | \( 1 - T + 2 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - T + 6 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 3 T + 2 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - T + 16 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 3 T + 18 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 7 T + 54 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $D_{4}$ | \( 1 + 9 T + 52 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 11 T + 82 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $D_{4}$ | \( 1 + 3 T + 58 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 7 T + 100 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 3 T + 122 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 7 T + 102 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 4 T - 42 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 89 | $D_{4}$ | \( 1 - 3 T + 28 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 15 T + 156 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
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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
−13.7755303384, −13.1370106999, −12.6701810859, −12.2507194527, −11.8900723896, −11.5144417519, −11.1492333219, −10.4472131809, −10.2791635058, −9.55727054225, −9.21776637310, −8.75854419870, −8.40987052646, −7.74772158612, −7.60608947087, −6.97485348008, −6.43954919301, −5.78563398750, −5.46871657327, −4.62053773212, −4.13118755458, −3.66167518036, −3.07906388336, −2.12798145931, −1.66933143878, 0,
1.66933143878, 2.12798145931, 3.07906388336, 3.66167518036, 4.13118755458, 4.62053773212, 5.46871657327, 5.78563398750, 6.43954919301, 6.97485348008, 7.60608947087, 7.74772158612, 8.40987052646, 8.75854419870, 9.21776637310, 9.55727054225, 10.2791635058, 10.4472131809, 11.1492333219, 11.5144417519, 11.8900723896, 12.2507194527, 12.6701810859, 13.1370106999, 13.7755303384
