| L(s) = 1 | + 2·2-s − 3-s + 2·4-s − 2·6-s − 4·7-s + 3·9-s + 9·11-s − 2·12-s − 8·14-s − 4·16-s + 6·18-s − 4·19-s + 4·21-s + 18·22-s + 6·25-s − 8·27-s − 8·28-s + 8·29-s − 8·32-s − 9·33-s + 6·36-s + 4·37-s − 8·38-s + 9·41-s + 8·42-s − 2·43-s + 18·44-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 0.577·3-s + 4-s − 0.816·6-s − 1.51·7-s + 9-s + 2.71·11-s − 0.577·12-s − 2.13·14-s − 16-s + 1.41·18-s − 0.917·19-s + 0.872·21-s + 3.83·22-s + 6/5·25-s − 1.53·27-s − 1.51·28-s + 1.48·29-s − 1.41·32-s − 1.56·33-s + 36-s + 0.657·37-s − 1.29·38-s + 1.40·41-s + 1.23·42-s − 0.304·43-s + 2.71·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 92416 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 92416 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.567679113\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.567679113\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - p T + p T^{2} \) |
| 19 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + T - 2 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 9 T + 38 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 40 T^{2} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 8 T + 66 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 12 T^{2} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 4 T + 10 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 9 T + 88 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 2 T + 34 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 4 T + 96 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 6 T + 58 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - T + 2 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 67 | $D_{4}$ | \( 1 + 5 T + 122 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 118 T^{2} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 11 T + 160 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 10 T + 30 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 13 T + 134 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 4 T - 42 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 3 T + 56 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
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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
−14.0068010220, −13.6480160639, −13.2278879869, −12.7714919452, −12.3518951734, −12.1831760851, −11.7562031593, −11.1430580944, −10.7996796728, −10.0878249300, −9.56360942667, −9.26523654423, −8.89423654658, −8.10649925393, −7.08284934108, −6.79967931371, −6.37905008529, −6.25029991790, −5.54044939656, −4.65147601966, −4.25159604071, −3.80522881891, −3.28430797567, −2.33983614990, −1.08330170597,
1.08330170597, 2.33983614990, 3.28430797567, 3.80522881891, 4.25159604071, 4.65147601966, 5.54044939656, 6.25029991790, 6.37905008529, 6.79967931371, 7.08284934108, 8.10649925393, 8.89423654658, 9.26523654423, 9.56360942667, 10.0878249300, 10.7996796728, 11.1430580944, 11.7562031593, 12.1831760851, 12.3518951734, 12.7714919452, 13.2278879869, 13.6480160639, 14.0068010220
