| L(s) = 1 | + 2-s + 2·3-s + 4-s + 2·6-s + 8-s + 3·9-s + 2·11-s + 2·12-s + 16-s − 4·17-s + 3·18-s + 2·22-s + 2·24-s + 6·25-s + 4·27-s + 32-s + 4·33-s − 4·34-s + 3·36-s + 4·41-s + 8·43-s + 2·44-s + 2·48-s − 10·49-s + 6·50-s − 8·51-s + 4·54-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 + 0.603·11-s + 0.577·12-s + 1/4·16-s − 0.970·17-s + 0.707·18-s + 0.426·22-s + 0.408·24-s + 6/5·25-s + 0.769·27-s + 0.176·32-s + 0.696·33-s − 0.685·34-s + 1/2·36-s + 0.624·41-s + 1.21·43-s + 0.301·44-s + 0.288·48-s − 1.42·49-s + 0.848·50-s − 1.12·51-s + 0.544·54-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 139392 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 139392 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.961392883\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.961392883\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 - T \) |
| 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| 11 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
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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
−9.316143739059841476691243080936, −8.822664775845247131815531688861, −8.492983922202548404765645833934, −7.72339303053252541131209387915, −7.50046630153661237864737028823, −6.79574406970657953900583400840, −6.46917181705411800265373292912, −5.88156384093744021354778323348, −5.11081223765090513639048258499, −4.39319022099111882503284652320, −4.25842178547984493307347569445, −3.34833436263269527102959669443, −2.90598744807222431902042213555, −2.20255326482175230591608211076, −1.34810073142484860857954126053,
1.34810073142484860857954126053, 2.20255326482175230591608211076, 2.90598744807222431902042213555, 3.34833436263269527102959669443, 4.25842178547984493307347569445, 4.39319022099111882503284652320, 5.11081223765090513639048258499, 5.88156384093744021354778323348, 6.46917181705411800265373292912, 6.79574406970657953900583400840, 7.50046630153661237864737028823, 7.72339303053252541131209387915, 8.492983922202548404765645833934, 8.822664775845247131815531688861, 9.316143739059841476691243080936
