| L(s) = 1 | + 4-s + 7-s − 3·13-s + 2·19-s − 25-s + 28-s + 2·37-s − 43-s − 3·52-s + 3·61-s − 64-s + 67-s + 3·73-s + 2·76-s + 79-s − 3·91-s − 100-s + 2·109-s − 2·121-s + 127-s + 131-s + 2·133-s + 137-s + 139-s + 2·148-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | + 4-s + 7-s − 3·13-s + 2·19-s − 25-s + 28-s + 2·37-s − 43-s − 3·52-s + 3·61-s − 64-s + 67-s + 3·73-s + 2·76-s + 79-s − 3·91-s − 100-s + 2·109-s − 2·121-s + 127-s + 131-s + 2·133-s + 137-s + 139-s + 2·148-s + 149-s + 151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1432809 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1432809 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.345487737\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.345487737\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 7 | $C_2$ | \( 1 - T + T^{2} \) |
| 19 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 5 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 13 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 + T + T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 29 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 31 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 43 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 59 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 61 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 - T + T^{2} ) \) |
| 67 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 73 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 - T + T^{2} ) \) |
| 79 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 89 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
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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
−10.01583888780973684443310417383, −9.860646636794211798176044900063, −9.387513015219020772231243313226, −9.186625490647177951874962885487, −8.280069389867319964318929317371, −7.86663099696123061960433942259, −7.75114370903899841959126221390, −7.34782318500529467945855327822, −6.92814618959255391384259315195, −6.57991421574174704681582379463, −5.99064213501844512452013705547, −5.25421775154805450742065349171, −5.03487898182613313204784085901, −4.92974864153073618594783409127, −4.05008969082018116296761560524, −3.55689175393236313144006706550, −2.71502677497380685916313096994, −2.36080899483554665605068810242, −2.08682203305425782209042157968, −1.07330832749316842843339695165,
1.07330832749316842843339695165, 2.08682203305425782209042157968, 2.36080899483554665605068810242, 2.71502677497380685916313096994, 3.55689175393236313144006706550, 4.05008969082018116296761560524, 4.92974864153073618594783409127, 5.03487898182613313204784085901, 5.25421775154805450742065349171, 5.99064213501844512452013705547, 6.57991421574174704681582379463, 6.92814618959255391384259315195, 7.34782318500529467945855327822, 7.75114370903899841959126221390, 7.86663099696123061960433942259, 8.280069389867319964318929317371, 9.186625490647177951874962885487, 9.387513015219020772231243313226, 9.860646636794211798176044900063, 10.01583888780973684443310417383
