| L(s) = 1 | + 8·4-s − 140·13-s − 56·19-s + 125·25-s − 308·31-s − 110·37-s − 1.04e3·43-s − 1.12e3·52-s − 182·61-s − 512·64-s + 880·67-s − 1.19e3·73-s − 448·76-s − 884·79-s − 2.66e3·97-s + 1.00e3·100-s − 1.82e3·103-s + 646·109-s + 1.33e3·121-s − 2.46e3·124-s + 127-s + 131-s + 137-s + 139-s − 880·148-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | + 4-s − 2.98·13-s − 0.676·19-s + 25-s − 1.78·31-s − 0.488·37-s − 3.68·43-s − 2.98·52-s − 0.382·61-s − 64-s + 1.60·67-s − 1.90·73-s − 0.676·76-s − 1.25·79-s − 2.78·97-s + 100-s − 1.74·103-s + 0.567·109-s + 121-s − 1.78·124-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s − 0.488·148-s + 0.000549·149-s + 0.000538·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 194481 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 194481 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.5455554896\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5455554896\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 - p^{3} T^{2} + p^{6} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - p^{3} T^{2} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - p^{3} T^{2} + p^{6} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 70 T + p^{3} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - p^{3} T^{2} + p^{6} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 107 T + p^{3} T^{2} )( 1 + 163 T + p^{3} T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - p^{3} T^{2} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 19 T + p^{3} T^{2} )( 1 + 289 T + p^{3} T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 323 T + p^{3} T^{2} )( 1 + 433 T + p^{3} T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 520 T + p^{3} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - p^{3} T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - p^{3} T^{2} + p^{6} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - p^{3} T^{2} + p^{6} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 719 T + p^{3} T^{2} )( 1 + 901 T + p^{3} T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 1007 T + p^{3} T^{2} )( 1 + 127 T + p^{3} T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 271 T + p^{3} T^{2} )( 1 + 919 T + p^{3} T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 503 T + p^{3} T^{2} )( 1 + 1387 T + p^{3} T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - p^{3} T^{2} + p^{6} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 1330 T + p^{3} T^{2} )^{2} \) |
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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
−11.30453194257185347936891286881, −10.27956083209803175970728823408, −10.25081169568286480137101051612, −9.567272421880884948901266038398, −9.332757193244271912365143230436, −8.529049672316125847966618328334, −8.251388036442077242208349159031, −7.45779498421284055779773259075, −7.22313989641582163104713892595, −6.70967973899054809091711450815, −6.64668630759271982649604512118, −5.41828436421701897590208947581, −5.39489699364415509355371136752, −4.65829605843546070449210196188, −4.20949255296742531605535240181, −3.05964471367929791532014632990, −2.92776921553246232766972661783, −1.94842047405757395210193268107, −1.80493315745799743006436770142, −0.21407224702063321874449893990,
0.21407224702063321874449893990, 1.80493315745799743006436770142, 1.94842047405757395210193268107, 2.92776921553246232766972661783, 3.05964471367929791532014632990, 4.20949255296742531605535240181, 4.65829605843546070449210196188, 5.39489699364415509355371136752, 5.41828436421701897590208947581, 6.64668630759271982649604512118, 6.70967973899054809091711450815, 7.22313989641582163104713892595, 7.45779498421284055779773259075, 8.251388036442077242208349159031, 8.529049672316125847966618328334, 9.332757193244271912365143230436, 9.567272421880884948901266038398, 10.25081169568286480137101051612, 10.27956083209803175970728823408, 11.30453194257185347936891286881
