| L(s) = 1 | − 2·2-s − 3-s + 2·4-s + 2·6-s + 9-s + 2·11-s − 2·12-s + 4·13-s − 4·16-s − 2·18-s − 4·22-s − 8·23-s − 25-s − 8·26-s − 27-s + 8·32-s − 2·33-s + 2·36-s − 4·39-s + 4·44-s + 16·46-s − 18·47-s + 4·48-s + 11·49-s + 2·50-s + 8·52-s + 2·54-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 0.577·3-s + 4-s + 0.816·6-s + 1/3·9-s + 0.603·11-s − 0.577·12-s + 1.10·13-s − 16-s − 0.471·18-s − 0.852·22-s − 1.66·23-s − 1/5·25-s − 1.56·26-s − 0.192·27-s + 1.41·32-s − 0.348·33-s + 1/3·36-s − 0.640·39-s + 0.603·44-s + 2.35·46-s − 2.62·47-s + 0.577·48-s + 11/7·49-s + 0.282·50-s + 1.10·52-s + 0.272·54-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 155952 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 155952 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + p T + p T^{2} \) |
| 3 | $C_1$ | \( 1 + T \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 5 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 11 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.836427381447004052087018681826, −8.742321760001961651389979974377, −8.160007849857414045140648086964, −7.47848778675296589128891353641, −7.43632371515983923998772054443, −6.41173993136661837405396202601, −6.29636417830616150950587212845, −5.80065514036351948381475791738, −4.91760568249510581017872290537, −4.32680757840779686690441622638, −3.81591641511052084615077615666, −2.94683125587980903660881898164, −1.81200784323266148009001357293, −1.34947676285250801422046885118, 0,
1.34947676285250801422046885118, 1.81200784323266148009001357293, 2.94683125587980903660881898164, 3.81591641511052084615077615666, 4.32680757840779686690441622638, 4.91760568249510581017872290537, 5.80065514036351948381475791738, 6.29636417830616150950587212845, 6.41173993136661837405396202601, 7.43632371515983923998772054443, 7.47848778675296589128891353641, 8.160007849857414045140648086964, 8.742321760001961651389979974377, 8.836427381447004052087018681826
