| L(s) = 1 | − 24·7-s − 124·17-s − 144·23-s + 186·25-s + 408·31-s + 44·41-s − 1.20e3·47-s − 254·49-s + 912·71-s + 1.64e3·73-s + 2.71e3·79-s + 1.87e3·89-s + 2.55e3·97-s − 1.89e3·103-s + 1.24e3·113-s + 2.97e3·119-s + 2.51e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 3.45e3·161-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | − 1.29·7-s − 1.76·17-s − 1.30·23-s + 1.48·25-s + 2.36·31-s + 0.167·41-s − 3.72·47-s − 0.740·49-s + 1.52·71-s + 2.63·73-s + 3.86·79-s + 2.23·89-s + 2.67·97-s − 1.81·103-s + 1.03·113-s + 2.29·119-s + 1.89·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 1.69·161-s + 0.000480·163-s + 0.000463·167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1327104 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1327104 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.713383257\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.713383257\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - 186 T^{2} + p^{6} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 12 T + p^{3} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 2518 T^{2} + p^{6} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 3994 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 62 T + p^{3} T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 2054 T^{2} + p^{6} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 72 T + p^{3} T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 32394 T^{2} + p^{6} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 204 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 49322 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 22 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 117398 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 600 T + p^{3} T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 232218 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 274826 T^{2} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 446906 T^{2} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 480422 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 456 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 822 T + p^{3} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 1356 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 1131910 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 938 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 1278 T + p^{3} T^{2} )^{2} \) |
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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
−9.537632404995594383477327956415, −9.372308067208420583702570582174, −8.857629011517218420482599704623, −8.343062224615406398777490456370, −7.995401160258820462855624880220, −7.78577175595520483713130097474, −6.82003486529962340353003755327, −6.49449998065306935175984411500, −6.39728605477763677591452819786, −6.31103761353169741421812367486, −5.06765661646327287209967904845, −5.01690214404273040329892142925, −4.52410848442112373346624290907, −3.86877645535031687897918520987, −3.28264206524264046864205554789, −3.08850997195347398361109585304, −2.16873219993995176586819503290, −2.02358474661958223288999036246, −0.840451307738406061285524141273, −0.40144566781329716112066346806,
0.40144566781329716112066346806, 0.840451307738406061285524141273, 2.02358474661958223288999036246, 2.16873219993995176586819503290, 3.08850997195347398361109585304, 3.28264206524264046864205554789, 3.86877645535031687897918520987, 4.52410848442112373346624290907, 5.01690214404273040329892142925, 5.06765661646327287209967904845, 6.31103761353169741421812367486, 6.39728605477763677591452819786, 6.49449998065306935175984411500, 6.82003486529962340353003755327, 7.78577175595520483713130097474, 7.995401160258820462855624880220, 8.343062224615406398777490456370, 8.857629011517218420482599704623, 9.372308067208420583702570582174, 9.537632404995594383477327956415
