| L(s) = 1 | + 3-s + 4-s − 2·7-s − 2·9-s + 12-s + 2·13-s + 16-s + 4·19-s − 2·21-s − 25-s − 5·27-s − 2·28-s − 8·31-s − 2·36-s − 14·37-s + 2·39-s − 2·43-s + 48-s − 11·49-s + 2·52-s + 4·57-s + 16·61-s + 4·63-s + 64-s + 28·67-s + 4·73-s − 75-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1/2·4-s − 0.755·7-s − 2/3·9-s + 0.288·12-s + 0.554·13-s + 1/4·16-s + 0.917·19-s − 0.436·21-s − 1/5·25-s − 0.962·27-s − 0.377·28-s − 1.43·31-s − 1/3·36-s − 2.30·37-s + 0.320·39-s − 0.304·43-s + 0.144·48-s − 1.57·49-s + 0.277·52-s + 0.529·57-s + 2.04·61-s + 0.503·63-s + 1/8·64-s + 3.42·67-s + 0.468·73-s − 0.115·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6084 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6084 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.035690323\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.035690323\) |
| \(L(\frac{3}{2})\) |
|
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$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 3 | $C_2$ | \( 1 - T + p T^{2} \) |
| 13 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
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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
−12.10810868275673738373780840946, −11.21486999262721520146961750481, −11.16559240112924374874969825279, −10.25096846899164676242515061975, −9.569105792671075621984918394603, −9.252086119982469784390910809299, −8.289125051595985253760732717123, −8.109212031498814193717580943346, −6.94019412385488865960361148937, −6.76423132327857768444846951818, −5.66417880764699758598698858020, −5.24164848449732929783448594193, −3.64028761626013442697591583469, −3.34467457020301574582723277384, −2.09859100780083292577684604843,
2.09859100780083292577684604843, 3.34467457020301574582723277384, 3.64028761626013442697591583469, 5.24164848449732929783448594193, 5.66417880764699758598698858020, 6.76423132327857768444846951818, 6.94019412385488865960361148937, 8.109212031498814193717580943346, 8.289125051595985253760732717123, 9.252086119982469784390910809299, 9.569105792671075621984918394603, 10.25096846899164676242515061975, 11.16559240112924374874969825279, 11.21486999262721520146961750481, 12.10810868275673738373780840946
