| L(s) = 1 | − 2·2-s − 3·3-s + 2·4-s − 2·5-s + 6·6-s − 7-s − 4·8-s + 4·9-s + 4·10-s − 3·11-s − 6·12-s − 3·13-s + 2·14-s + 6·15-s + 8·16-s + 3·17-s − 8·18-s + 5·19-s − 4·20-s + 3·21-s + 6·22-s − 3·23-s + 12·24-s + 2·25-s + 6·26-s − 6·27-s − 2·28-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1.73·3-s + 4-s − 0.894·5-s + 2.44·6-s − 0.377·7-s − 1.41·8-s + 4/3·9-s + 1.26·10-s − 0.904·11-s − 1.73·12-s − 0.832·13-s + 0.534·14-s + 1.54·15-s + 2·16-s + 0.727·17-s − 1.88·18-s + 1.14·19-s − 0.894·20-s + 0.654·21-s + 1.27·22-s − 0.625·23-s + 2.44·24-s + 2/5·25-s + 1.17·26-s − 1.15·27-s − 0.377·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1403 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1403 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 23 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 2 T + p T^{2} ) \) |
| 61 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 6 T + p T^{2} ) \) |
| good | 2 | $C_2^2$ | \( 1 + p T + p T^{2} + p^{2} T^{3} + p^{2} T^{4} \) |
| 3 | $C_4$ | \( 1 + p T + 5 T^{2} + p^{2} T^{3} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 7 | $D_{4}$ | \( 1 + T + p T^{2} + p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 3 T + 15 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 3 T + 21 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 3 T + 35 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 5 T + 22 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $D_{4}$ | \( 1 + 8 T + 50 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + T - 56 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 4 T + 10 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 3 T + 57 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - T + 18 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - T + 13 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 7 T + 32 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 9 T + 106 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 2 T + 30 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 6 T + 66 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 3 T - 3 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 17 T + 183 T^{2} - 17 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + T + 18 T^{2} + p T^{3} + p^{2} T^{4} \) |
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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
−19.3511590000, −19.0280637683, −18.3365526062, −18.0470193680, −17.7126558399, −16.8509857956, −16.7002367046, −16.1298215250, −15.5338339458, −15.0925819871, −14.2368020772, −13.1290129319, −12.2983097321, −12.0533221795, −11.5672886058, −10.8650909748, −10.2321024076, −9.69221054347, −9.00337710612, −7.84992525454, −7.61663888445, −6.56642311068, −5.70429225088, −5.08881624682, −3.33741085275, 0,
3.33741085275, 5.08881624682, 5.70429225088, 6.56642311068, 7.61663888445, 7.84992525454, 9.00337710612, 9.69221054347, 10.2321024076, 10.8650909748, 11.5672886058, 12.0533221795, 12.2983097321, 13.1290129319, 14.2368020772, 15.0925819871, 15.5338339458, 16.1298215250, 16.7002367046, 16.8509857956, 17.7126558399, 18.0470193680, 18.3365526062, 19.0280637683, 19.3511590000
