| L(s) = 1 | − 2·2-s + 2·3-s + 3·4-s − 6·5-s − 4·6-s + 4·7-s − 4·8-s + 3·9-s + 12·10-s − 2·11-s + 6·12-s + 6·13-s − 8·14-s − 12·15-s + 5·16-s − 6·18-s + 8·19-s − 18·20-s + 8·21-s + 4·22-s − 8·24-s + 19·25-s − 12·26-s + 4·27-s + 12·28-s − 6·29-s + 24·30-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 1.15·3-s + 3/2·4-s − 2.68·5-s − 1.63·6-s + 1.51·7-s − 1.41·8-s + 9-s + 3.79·10-s − 0.603·11-s + 1.73·12-s + 1.66·13-s − 2.13·14-s − 3.09·15-s + 5/4·16-s − 1.41·18-s + 1.83·19-s − 4.02·20-s + 1.74·21-s + 0.852·22-s − 1.63·24-s + 19/5·25-s − 2.35·26-s + 0.769·27-s + 2.26·28-s − 1.11·29-s + 4.38·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16208676 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16208676 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.985366498\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.985366498\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 11 | $C_1$ | \( ( 1 + T )^{2} \) |
| 61 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 6 T + 17 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 8 T + 46 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 6 T + 35 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 10 T + 85 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 2 T - 15 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 53 | $D_{4}$ | \( 1 - 16 T + 162 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 6 T - T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 73 | $D_{4}$ | \( 1 - 8 T + 130 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $D_{4}$ | \( 1 - 12 T + 74 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 6 T + 115 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 10 T + 187 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.314291976578453610425888977880, −8.229507591756736484738951426170, −8.078760550304210497098668502497, −7.896813689785854735339645540580, −7.35767276084291299501516984542, −7.20189899092246941679534621455, −6.70362464417814807820538617865, −6.43459416827833417170287181052, −5.49338989545799834085282449193, −5.32582134653663642260418831195, −4.56612445392026760303154903145, −4.53536886038287770044526025115, −3.71156247084677365531160923126, −3.58842894662310630317536590998, −3.18142311305950756207056981513, −2.89495616777109967258441212075, −1.84728417562563442790061756854, −1.78513181048759637198992656317, −0.832561977175370040387388497229, −0.68512084179758343425646155118,
0.68512084179758343425646155118, 0.832561977175370040387388497229, 1.78513181048759637198992656317, 1.84728417562563442790061756854, 2.89495616777109967258441212075, 3.18142311305950756207056981513, 3.58842894662310630317536590998, 3.71156247084677365531160923126, 4.53536886038287770044526025115, 4.56612445392026760303154903145, 5.32582134653663642260418831195, 5.49338989545799834085282449193, 6.43459416827833417170287181052, 6.70362464417814807820538617865, 7.20189899092246941679534621455, 7.35767276084291299501516984542, 7.896813689785854735339645540580, 8.078760550304210497098668502497, 8.229507591756736484738951426170, 8.314291976578453610425888977880
