| L(s) = 1 | − 2-s − 2·3-s + 4-s − 4·5-s + 2·6-s + 7-s + 8-s + 2·9-s + 4·10-s + 2·11-s − 2·12-s − 2·13-s − 14-s + 8·15-s − 3·16-s − 2·17-s − 2·18-s + 19-s − 4·20-s − 2·21-s − 2·22-s + 5·23-s − 2·24-s + 6·25-s + 2·26-s − 6·27-s + 28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.15·3-s + 1/2·4-s − 1.78·5-s + 0.816·6-s + 0.377·7-s + 0.353·8-s + 2/3·9-s + 1.26·10-s + 0.603·11-s − 0.577·12-s − 0.554·13-s − 0.267·14-s + 2.06·15-s − 3/4·16-s − 0.485·17-s − 0.471·18-s + 0.229·19-s − 0.894·20-s − 0.436·21-s − 0.426·22-s + 1.04·23-s − 0.408·24-s + 6/5·25-s + 0.392·26-s − 1.15·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 394 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 394 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2007827436\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2007827436\) |
| \(L(\frac{3}{2})\) |
|
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_1$$\times$$C_2$ | \( ( 1 - T )( 1 + p T + p T^{2} ) \) |
| 197 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - 18 T + p T^{2} ) \) |
| good | 3 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 7 | $D_{4}$ | \( 1 - T - 2 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 2 T + 22 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 2 T - 6 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - T - 22 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 5 T + 30 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 3 T + 28 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 6 T + 30 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 3 T + 44 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - T - 6 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + p T^{2} ) \) |
| 61 | $D_{4}$ | \( 1 + T + 16 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $D_{4}$ | \( 1 - 2 T + 102 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 2 T - 62 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 42 T^{2} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 9 T + 66 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
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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
−19.6295715505, −19.3059469824, −19.0248356528, −17.9947296559, −17.5176256677, −16.8487865706, −16.5231674940, −15.6004576954, −15.4443043074, −14.5647329690, −13.5078710086, −12.5489303651, −11.7483015383, −11.4806467239, −10.9979364628, −10.0972223242, −9.01903323173, −8.03781843767, −7.36037006983, −6.65135433456, −5.09410049950, −4.06198103368,
4.06198103368, 5.09410049950, 6.65135433456, 7.36037006983, 8.03781843767, 9.01903323173, 10.0972223242, 10.9979364628, 11.4806467239, 11.7483015383, 12.5489303651, 13.5078710086, 14.5647329690, 15.4443043074, 15.6004576954, 16.5231674940, 16.8487865706, 17.5176256677, 17.9947296559, 19.0248356528, 19.3059469824, 19.6295715505
