| L(s) = 1 | − 4-s + 2·7-s + 2·13-s − 3·16-s + 3·19-s − 2·28-s + 10·31-s − 4·37-s − 4·43-s − 2·49-s − 2·52-s − 8·61-s + 7·64-s + 2·67-s − 4·73-s − 3·76-s + 4·79-s + 4·91-s + 14·97-s + 8·103-s − 14·109-s − 6·112-s + 14·121-s − 10·124-s + 127-s + 131-s + 6·133-s + ⋯ |
| L(s) = 1 | − 1/2·4-s + 0.755·7-s + 0.554·13-s − 3/4·16-s + 0.688·19-s − 0.377·28-s + 1.79·31-s − 0.657·37-s − 0.609·43-s − 2/7·49-s − 0.277·52-s − 1.02·61-s + 7/8·64-s + 0.244·67-s − 0.468·73-s − 0.344·76-s + 0.450·79-s + 0.419·91-s + 1.42·97-s + 0.788·103-s − 1.34·109-s − 0.566·112-s + 1.27·121-s − 0.898·124-s + 0.0887·127-s + 0.0873·131-s + 0.520·133-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 961875 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 961875 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.879716818\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.879716818\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| 19 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - 2 T + p T^{2} ) \) |
| good | 2 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 20 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 70 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 86 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
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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
−8.191063267747204379246728812215, −7.79299109407548105183651796404, −7.34081177954816747224363878474, −6.75495896163846866268716243282, −6.44506554897564806369745191099, −5.89865135636990142636287836411, −5.35321516780934268651586553298, −4.90169586335377487963386503725, −4.52740816792627750380325032619, −4.11883635151693151676754510789, −3.37280973143547339449146795377, −2.96647391623180758430622930866, −2.14148034080307729172786887827, −1.52644445037254599624864137489, −0.67335071423937653134473085898,
0.67335071423937653134473085898, 1.52644445037254599624864137489, 2.14148034080307729172786887827, 2.96647391623180758430622930866, 3.37280973143547339449146795377, 4.11883635151693151676754510789, 4.52740816792627750380325032619, 4.90169586335377487963386503725, 5.35321516780934268651586553298, 5.89865135636990142636287836411, 6.44506554897564806369745191099, 6.75495896163846866268716243282, 7.34081177954816747224363878474, 7.79299109407548105183651796404, 8.191063267747204379246728812215
