| L(s) = 1 | + 2-s − 3·3-s + 4-s − 5-s − 3·6-s − 7-s + 8-s + 2·9-s − 10-s − 3·12-s + 13-s − 14-s + 3·15-s + 16-s + 17-s + 2·18-s + 19-s − 20-s + 3·21-s − 2·23-s − 3·24-s + 4·25-s + 26-s + 6·27-s − 28-s − 4·29-s + 3·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.73·3-s + 1/2·4-s − 0.447·5-s − 1.22·6-s − 0.377·7-s + 0.353·8-s + 2/3·9-s − 0.316·10-s − 0.866·12-s + 0.277·13-s − 0.267·14-s + 0.774·15-s + 1/4·16-s + 0.242·17-s + 0.471·18-s + 0.229·19-s − 0.223·20-s + 0.654·21-s − 0.417·23-s − 0.612·24-s + 4/5·25-s + 0.196·26-s + 1.15·27-s − 0.188·28-s − 0.742·29-s + 0.547·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 976 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 976 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4448401744\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4448401744\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 - T \) |
| 61 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - 4 T + p T^{2} ) \) |
| good | 3 | $D_{4}$ | \( 1 + p T + 7 T^{2} + p^{2} T^{3} + p^{2} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + T - 3 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + T + p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - T - 11 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - T + 18 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - T + 6 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 4 T + 24 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 10 T + 54 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 12 T + 88 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - T + 71 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 5 T + 18 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 3 T + 82 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 4 T + 48 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 9 T + 38 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 8 T + 70 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 73 | $D_{4}$ | \( 1 - 11 T + 124 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 6 T + 96 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 15 T + 170 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 11 T + 194 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 3 T + 69 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
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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
−19.4586025370, −18.9749133632, −18.2304605358, −17.6733498957, −17.0742839009, −16.6817637323, −16.1533018451, −15.5877137378, −15.0321617582, −14.1137911686, −13.6578656160, −12.7490174834, −12.1505566654, −11.8317748852, −11.1890472479, −10.6491298476, −9.97417646363, −8.80177061714, −7.87057358791, −6.76763269606, −6.25568035712, −5.43141387719, −4.72869776713, −3.34642751055,
3.34642751055, 4.72869776713, 5.43141387719, 6.25568035712, 6.76763269606, 7.87057358791, 8.80177061714, 9.97417646363, 10.6491298476, 11.1890472479, 11.8317748852, 12.1505566654, 12.7490174834, 13.6578656160, 14.1137911686, 15.0321617582, 15.5877137378, 16.1533018451, 16.6817637323, 17.0742839009, 17.6733498957, 18.2304605358, 18.9749133632, 19.4586025370
