| L(s) = 1 | + 4·3-s + 6·9-s − 4·11-s − 4·17-s + 4·19-s − 6·25-s − 4·27-s − 16·33-s − 12·41-s + 12·43-s + 2·49-s − 16·51-s + 16·57-s + 28·59-s + 20·67-s + 28·73-s − 24·75-s − 37·81-s − 12·83-s − 4·89-s − 4·97-s − 24·99-s − 4·107-s + 4·113-s − 10·121-s − 48·123-s + 127-s + ⋯ |
| L(s) = 1 | + 2.30·3-s + 2·9-s − 1.20·11-s − 0.970·17-s + 0.917·19-s − 6/5·25-s − 0.769·27-s − 2.78·33-s − 1.87·41-s + 1.82·43-s + 2/7·49-s − 2.24·51-s + 2.11·57-s + 3.64·59-s + 2.44·67-s + 3.27·73-s − 2.77·75-s − 4.11·81-s − 1.31·83-s − 0.423·89-s − 0.406·97-s − 2.41·99-s − 0.386·107-s + 0.376·113-s − 0.909·121-s − 4.32·123-s + 0.0887·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16384 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16384 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.960380722\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.960380722\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| good | 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 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
−11.15482415732823435746426270865, −10.22941768705266383849925633731, −9.742591354048855107896325616652, −9.385355573475711088007535446099, −8.728992519223720851294830795286, −8.173183686627627779120693205031, −8.098681382024412731658064042949, −7.33738117792255002879146989083, −6.78571919296342801026858814862, −5.64130249043000575312616744005, −5.16542238567969836254694300670, −3.91058908136761919842529082491, −3.56276783709589056726776122484, −2.46608091328452001690059553408, −2.30676446591740616355718333113,
2.30676446591740616355718333113, 2.46608091328452001690059553408, 3.56276783709589056726776122484, 3.91058908136761919842529082491, 5.16542238567969836254694300670, 5.64130249043000575312616744005, 6.78571919296342801026858814862, 7.33738117792255002879146989083, 8.098681382024412731658064042949, 8.173183686627627779120693205031, 8.728992519223720851294830795286, 9.385355573475711088007535446099, 9.742591354048855107896325616652, 10.22941768705266383849925633731, 11.15482415732823435746426270865
