| L(s) = 1 | − 3-s + 4-s − 2·5-s − 3·7-s − 12-s + 13-s + 2·15-s + 16-s + 3·19-s − 2·20-s + 3·21-s − 2·23-s + 3·25-s + 27-s − 3·28-s − 6·31-s + 6·35-s + 3·37-s − 39-s + 14·41-s − 10·43-s + 2·47-s − 48-s + 5·49-s + 52-s − 4·53-s − 3·57-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/2·4-s − 0.894·5-s − 1.13·7-s − 0.288·12-s + 0.277·13-s + 0.516·15-s + 1/4·16-s + 0.688·19-s − 0.447·20-s + 0.654·21-s − 0.417·23-s + 3/5·25-s + 0.192·27-s − 0.566·28-s − 1.07·31-s + 1.01·35-s + 0.493·37-s − 0.160·39-s + 2.18·41-s − 1.52·43-s + 0.291·47-s − 0.144·48-s + 5/7·49-s + 0.138·52-s − 0.549·53-s − 0.397·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1116 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1116 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4354897184\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4354897184\) |
| \(L(\frac{3}{2})\) |
|
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_1$ | \( ( 1 - T )( 1 + T ) \) |
| 3 | $C_2$ | \( 1 + T + T^{2} \) |
| 31 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 7 T + p T^{2} ) \) |
| good | 5 | $D_{4}$ | \( 1 + 2 T + T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - T - p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $D_{4}$ | \( 1 + 2 T + 7 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 16 T^{2} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 3 T - 8 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 + 10 T + 50 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $D_{4}$ | \( 1 - 10 T + 82 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 3 T + 70 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 71 | $D_{4}$ | \( 1 + 12 T + 100 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + T - 22 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 7 T + 62 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 89 | $D_{4}$ | \( 1 + 8 T + 16 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 6 T + 142 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
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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
−19.8269873343, −19.2883757539, −18.8516756234, −18.0948767073, −17.7136828040, −16.6192987778, −16.5175014031, −15.9517998857, −15.5242386030, −14.7983059252, −14.1647667290, −13.2277394012, −12.7370138563, −12.1259293035, −11.4968156485, −11.0450555869, −10.2451872085, −9.55019977604, −8.71792977509, −7.72875675432, −7.10805164474, −6.27604807303, −5.52797435783, −4.15236732343, −3.09412679935,
3.09412679935, 4.15236732343, 5.52797435783, 6.27604807303, 7.10805164474, 7.72875675432, 8.71792977509, 9.55019977604, 10.2451872085, 11.0450555869, 11.4968156485, 12.1259293035, 12.7370138563, 13.2277394012, 14.1647667290, 14.7983059252, 15.5242386030, 15.9517998857, 16.5175014031, 16.6192987778, 17.7136828040, 18.0948767073, 18.8516756234, 19.2883757539, 19.8269873343
