| L(s) = 1 | + 7-s + 13-s − 2·19-s − 25-s − 2·31-s − 2·37-s − 2·43-s + 49-s + 61-s + 67-s − 2·73-s + 79-s + 91-s + 97-s + 103-s + 4·109-s − 121-s + 127-s + 131-s − 2·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | + 7-s + 13-s − 2·19-s − 25-s − 2·31-s − 2·37-s − 2·43-s + 49-s + 61-s + 67-s − 2·73-s + 79-s + 91-s + 97-s + 103-s + 4·109-s − 121-s + 127-s + 131-s − 2·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 104976 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 104976 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6657956224\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6657956224\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 7 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 13 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 19 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 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_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 73 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 79 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 97 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
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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
−11.97570676120490307172917231320, −11.53224737472621219122693180508, −11.05557552698172634847856086338, −10.87358085027830755818817154164, −10.17597287541905763716699842238, −10.00178985013388440685703518763, −9.012409442744059994790052722607, −8.756247128826566459550341331090, −8.383230055444825746770459929583, −7.978273272407557497593735088428, −7.06371752062564521444756463495, −7.04845812006168297162255011282, −6.00371873638739734355633250784, −5.87782717040273031102128602937, −4.99972171567517710473356813659, −4.62038351773339867976712958532, −3.70377076016972246204300146266, −3.53451938798345923153441671203, −2.04999548292918478838892789240, −1.80763934899282826814115515555,
1.80763934899282826814115515555, 2.04999548292918478838892789240, 3.53451938798345923153441671203, 3.70377076016972246204300146266, 4.62038351773339867976712958532, 4.99972171567517710473356813659, 5.87782717040273031102128602937, 6.00371873638739734355633250784, 7.04845812006168297162255011282, 7.06371752062564521444756463495, 7.978273272407557497593735088428, 8.383230055444825746770459929583, 8.756247128826566459550341331090, 9.012409442744059994790052722607, 10.00178985013388440685703518763, 10.17597287541905763716699842238, 10.87358085027830755818817154164, 11.05557552698172634847856086338, 11.53224737472621219122693180508, 11.97570676120490307172917231320
