| L(s) = 1 | − 2-s + 4-s − 8·7-s − 8-s − 6·9-s + 4·11-s − 4·13-s + 8·14-s + 16-s + 6·18-s − 4·19-s − 4·22-s + 23-s + 6·25-s + 4·26-s − 8·28-s + 4·29-s − 32-s − 6·36-s + 4·38-s + 12·41-s + 20·43-s + 4·44-s − 46-s + 34·49-s − 6·50-s − 4·52-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 3.02·7-s − 0.353·8-s − 2·9-s + 1.20·11-s − 1.10·13-s + 2.13·14-s + 1/4·16-s + 1.41·18-s − 0.917·19-s − 0.852·22-s + 0.208·23-s + 6/5·25-s + 0.784·26-s − 1.51·28-s + 0.742·29-s − 0.176·32-s − 36-s + 0.648·38-s + 1.87·41-s + 3.04·43-s + 0.603·44-s − 0.147·46-s + 34/7·49-s − 0.848·50-s − 0.554·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 389344 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 389344 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | $C_1$ | \( 1 + T \) |
| 23 | $C_1$ | \( 1 - T \) |
| good | 3 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
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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
−8.900141268185548824896487898780, −7.934282519903949710148233882009, −7.58296953347332318996315151914, −6.74207540337883817408246132970, −6.69441913641732873591204403077, −6.23765066001608948536504848026, −5.84432417959727908710252168629, −5.40734526116830777226298884981, −4.34183165372739795918268010061, −3.88700502119652044270802644739, −3.10158434946020981235655663049, −2.63232737474552247848840602025, −2.63067144427293189510153997558, −0.807357615430292443567249537652, 0,
0.807357615430292443567249537652, 2.63067144427293189510153997558, 2.63232737474552247848840602025, 3.10158434946020981235655663049, 3.88700502119652044270802644739, 4.34183165372739795918268010061, 5.40734526116830777226298884981, 5.84432417959727908710252168629, 6.23765066001608948536504848026, 6.69441913641732873591204403077, 6.74207540337883817408246132970, 7.58296953347332318996315151914, 7.934282519903949710148233882009, 8.900141268185548824896487898780
