L(s) = 1 | + 2-s + 2·3-s + 4-s + 2·6-s + 8-s − 3·9-s − 6·11-s + 2·12-s + 16-s − 3·18-s + 4·19-s − 6·22-s + 2·24-s − 10·25-s − 14·27-s + 32-s − 12·33-s − 3·36-s + 4·38-s + 6·41-s + 16·43-s − 6·44-s + 2·48-s + 49-s − 10·50-s − 14·54-s + 8·57-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1.15·3-s + 1/2·4-s + 0.816·6-s + 0.353·8-s − 9-s − 1.80·11-s + 0.577·12-s + 1/4·16-s − 0.707·18-s + 0.917·19-s − 1.27·22-s + 0.408·24-s − 2·25-s − 2.69·27-s + 0.176·32-s − 2.08·33-s − 1/2·36-s + 0.648·38-s + 0.937·41-s + 2.43·43-s − 0.904·44-s + 0.288·48-s + 1/7·49-s − 1.41·50-s − 1.90·54-s + 1.05·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1059968 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1059968 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.263772781\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.263772781\) |
\(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 - T \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
good | 3 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 18 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 17 T + p T^{2} )^{2} \) |
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
−7.974204239601213188956236456444, −7.63380170955425435565378568582, −7.60007770624079585422104671113, −6.81334322707606658549650565225, −6.10743185313712551583625613621, −5.71034209994038661023746867842, −5.49329025649653204207581393927, −5.06509234546123458101387314677, −4.34218533360235296422646954393, −3.60858467418748472558474969025, −3.54211448492338401696398190009, −2.72564141720431011383752495662, −2.33000973337707432224237511883, −2.17542049731475021336825361662, −0.63890669930845811510933037152,
0.63890669930845811510933037152, 2.17542049731475021336825361662, 2.33000973337707432224237511883, 2.72564141720431011383752495662, 3.54211448492338401696398190009, 3.60858467418748472558474969025, 4.34218533360235296422646954393, 5.06509234546123458101387314677, 5.49329025649653204207581393927, 5.71034209994038661023746867842, 6.10743185313712551583625613621, 6.81334322707606658549650565225, 7.60007770624079585422104671113, 7.63380170955425435565378568582, 7.974204239601213188956236456444