| L(s) = 1 | − 2·2-s − 3-s + 4-s − 2·5-s + 2·6-s + 7-s + 9-s + 4·10-s + 2·11-s − 12-s − 2·13-s − 2·14-s + 2·15-s + 16-s − 2·18-s + 19-s − 2·20-s − 21-s − 4·22-s + 2·23-s − 2·25-s + 4·26-s − 4·27-s + 28-s − 9·29-s − 4·30-s + 4·31-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 0.577·3-s + 1/2·4-s − 0.894·5-s + 0.816·6-s + 0.377·7-s + 1/3·9-s + 1.26·10-s + 0.603·11-s − 0.288·12-s − 0.554·13-s − 0.534·14-s + 0.516·15-s + 1/4·16-s − 0.471·18-s + 0.229·19-s − 0.447·20-s − 0.218·21-s − 0.852·22-s + 0.417·23-s − 2/5·25-s + 0.784·26-s − 0.769·27-s + 0.188·28-s − 1.67·29-s − 0.730·30-s + 0.718·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 295 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 295 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1492683688\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1492683688\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + T + p T^{2} ) \) |
| 59 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 4 T + p T^{2} ) \) |
| good | 2 | $D_{4}$ | \( 1 + p T + 3 T^{2} + p^{2} T^{3} + p^{2} T^{4} \) |
| 3 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 7 | $D_{4}$ | \( 1 - T - p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $D_{4}$ | \( 1 + 2 T - 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - T - 6 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 2 T - 6 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 9 T + 64 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 4 T + 6 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 2 T + 38 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - T + 68 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 2 T + 22 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 10 T + 90 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 9 T + 112 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 102 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 - 66 T^{2} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - T + 110 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 14 T + 138 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 138 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−19.5943790917, −19.1705597141, −18.6253981472, −18.1063434879, −17.3428767194, −17.0651471551, −16.4697340363, −15.4409573542, −15.1670527978, −14.1846862585, −13.2469970219, −12.2291821794, −11.6895301751, −11.0785083624, −10.0882154598, −9.41053737416, −8.65224011130, −7.78228122162, −7.06932004114, −5.61010872777, −4.12033228144,
4.12033228144, 5.61010872777, 7.06932004114, 7.78228122162, 8.65224011130, 9.41053737416, 10.0882154598, 11.0785083624, 11.6895301751, 12.2291821794, 13.2469970219, 14.1846862585, 15.1670527978, 15.4409573542, 16.4697340363, 17.0651471551, 17.3428767194, 18.1063434879, 18.6253981472, 19.1705597141, 19.5943790917
