| L(s) = 1 | + 2·2-s + 2·3-s + 3·4-s − 2·5-s + 4·6-s − 7-s + 4·8-s + 3·9-s − 4·10-s + 5·11-s + 6·12-s + 12·13-s − 2·14-s − 4·15-s + 5·16-s − 8·17-s + 6·18-s + 19-s − 6·20-s − 2·21-s + 10·22-s + 5·23-s + 8·24-s + 3·25-s + 24·26-s + 4·27-s − 3·28-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 1.15·3-s + 3/2·4-s − 0.894·5-s + 1.63·6-s − 0.377·7-s + 1.41·8-s + 9-s − 1.26·10-s + 1.50·11-s + 1.73·12-s + 3.32·13-s − 0.534·14-s − 1.03·15-s + 5/4·16-s − 1.94·17-s + 1.41·18-s + 0.229·19-s − 1.34·20-s − 0.436·21-s + 2.13·22-s + 1.04·23-s + 1.63·24-s + 3/5·25-s + 4.70·26-s + 0.769·27-s − 0.566·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 864900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(8.717567198\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.717567198\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 31 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 7 | $D_{4}$ | \( 1 + T - 2 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 5 T + 12 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 19 | $D_{4}$ | \( 1 - T + 22 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 5 T + 36 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $D_{4}$ | \( 1 - 2 T + 10 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 2 T + 18 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 9 T + 90 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 6 T + 38 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 3 T + 92 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 2 T + 54 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 6 T + 66 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 6 T + 78 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 17 T + 198 T^{2} + 17 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 9 T + 150 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 9 T + 162 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 89 | $D_{4}$ | \( 1 + 5 T + 168 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 66 T^{2} + p^{2} T^{4} \) |
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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
−10.38493600913261989215416619688, −9.961760495444729164998456053953, −9.074285732843307171420637346452, −8.878026530850102149779363867399, −8.691787669625172274776889988903, −8.358868109108267568748612311827, −7.66018531301354046553157548275, −7.14162691728828851538160210390, −6.64973642707212625780897949381, −6.60988606686902975734274433061, −5.97098794366318272387691963461, −5.62931002771655339073818882117, −4.47451138677009031822657834858, −4.41488377043327344288914559931, −3.97913276916945141287673413797, −3.52424350148340957764605013347, −3.17058699423702372500460411658, −2.65989475046172098984043666958, −1.52002144829545026651106709851, −1.31343505123261870533423959538,
1.31343505123261870533423959538, 1.52002144829545026651106709851, 2.65989475046172098984043666958, 3.17058699423702372500460411658, 3.52424350148340957764605013347, 3.97913276916945141287673413797, 4.41488377043327344288914559931, 4.47451138677009031822657834858, 5.62931002771655339073818882117, 5.97098794366318272387691963461, 6.60988606686902975734274433061, 6.64973642707212625780897949381, 7.14162691728828851538160210390, 7.66018531301354046553157548275, 8.358868109108267568748612311827, 8.691787669625172274776889988903, 8.878026530850102149779363867399, 9.074285732843307171420637346452, 9.961760495444729164998456053953, 10.38493600913261989215416619688
