| L(s) = 1 | − 2·3-s + 2·5-s − 3·7-s − 11-s + 13-s − 4·15-s − 4·16-s + 2·17-s − 19-s + 6·21-s + 4·23-s − 2·25-s + 5·27-s + 6·29-s − 4·31-s + 2·33-s − 6·35-s + 9·37-s − 2·39-s + 6·41-s + 2·43-s + 2·47-s + 8·48-s + 3·49-s − 4·51-s − 5·53-s − 2·55-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.894·5-s − 1.13·7-s − 0.301·11-s + 0.277·13-s − 1.03·15-s − 16-s + 0.485·17-s − 0.229·19-s + 1.30·21-s + 0.834·23-s − 2/5·25-s + 0.962·27-s + 1.11·29-s − 0.718·31-s + 0.348·33-s − 1.01·35-s + 1.47·37-s − 0.320·39-s + 0.937·41-s + 0.304·43-s + 0.291·47-s + 1.15·48-s + 3/7·49-s − 0.560·51-s − 0.686·53-s − 0.269·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1077 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1077 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4062914313\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4062914313\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + T + p T^{2} ) \) |
| 359 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 30 T + p T^{2} ) \) |
| good | 2 | $C_2$ | \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 5 | $D_{4}$ | \( 1 - 2 T + 6 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 3 T + 6 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $D_{4}$ | \( 1 - 2 T + 6 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 23 | $D_{4}$ | \( 1 - 4 T + 36 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) |
| 31 | $D_{4}$ | \( 1 + 4 T + 10 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 9 T + 47 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 6 T + 70 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 2 T - 52 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 53 | $D_{4}$ | \( 1 + 5 T + 75 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 3 T - 22 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 14 T + 160 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_4$ | \( 1 + 6 T + 26 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 3 T - 27 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 3 T + 78 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 10 T + 90 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 5 T + 3 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - T - 12 T^{2} - p T^{3} + p^{2} T^{4} \) |
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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
−19.7885707311, −19.3350967233, −18.6069266154, −18.0063475734, −17.7391719162, −17.0104791184, −16.6066332273, −16.1033269463, −15.5801961080, −14.7151030466, −13.9721229615, −13.4012859044, −12.8795356592, −12.2404190603, −11.5432279366, −10.8230715911, −10.3922726230, −9.40796104497, −9.14344422481, −7.87261251070, −6.77845005959, −6.13876455308, −5.67161544690, −4.56188709982, −2.87034808619,
2.87034808619, 4.56188709982, 5.67161544690, 6.13876455308, 6.77845005959, 7.87261251070, 9.14344422481, 9.40796104497, 10.3922726230, 10.8230715911, 11.5432279366, 12.2404190603, 12.8795356592, 13.4012859044, 13.9721229615, 14.7151030466, 15.5801961080, 16.1033269463, 16.6066332273, 17.0104791184, 17.7391719162, 18.0063475734, 18.6069266154, 19.3350967233, 19.7885707311
