| L(s) = 1 | + 4-s − 7-s + 3·19-s − 28-s + 37-s + 4·43-s − 3·61-s − 64-s − 67-s − 3·73-s + 3·76-s + 79-s − 3·103-s − 109-s + 121-s + 127-s + 131-s − 3·133-s + 137-s + 139-s + 148-s + 149-s + 151-s + 157-s + 163-s + 167-s − 169-s + ⋯ |
| L(s) = 1 | + 4-s − 7-s + 3·19-s − 28-s + 37-s + 4·43-s − 3·61-s − 64-s − 67-s − 3·73-s + 3·76-s + 79-s − 3·103-s − 109-s + 121-s + 127-s + 131-s − 3·133-s + 137-s + 139-s + 148-s + 149-s + 151-s + 157-s + 163-s + 167-s − 169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2480625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2480625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.410286689\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.410286689\) |
| \(L(1)\) |
|
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 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + T + T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 11 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 13 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 19 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 - T + T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 29 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 37 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 43 | $C_1$ | \( ( 1 - T )^{4} \) |
| 47 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 59 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 61 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 + T + T^{2} ) \) |
| 67 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 73 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 + T + T^{2} ) \) |
| 79 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 89 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
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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
−9.571165917370971600287195208813, −9.541884241741345333064276369543, −9.106073597725074568894866773615, −8.893357507536551178093015173587, −7.87874864765669193479236167158, −7.79488126140241289617004408679, −7.37919937047148341501953087265, −7.10070670875801345571919352382, −6.69253119144104813150899422309, −5.98100840600319990675127526828, −5.85900599019417237478912698898, −5.63389040558050400263991941961, −4.76475524543023702655972934511, −4.41956878938508141687838840019, −3.82677478317679182518084185679, −3.12760345105075596315641598121, −2.88104399043898628402685523472, −2.59511306074975866219956695347, −1.60655495480169230664533452451, −1.02460460141546533731090149360,
1.02460460141546533731090149360, 1.60655495480169230664533452451, 2.59511306074975866219956695347, 2.88104399043898628402685523472, 3.12760345105075596315641598121, 3.82677478317679182518084185679, 4.41956878938508141687838840019, 4.76475524543023702655972934511, 5.63389040558050400263991941961, 5.85900599019417237478912698898, 5.98100840600319990675127526828, 6.69253119144104813150899422309, 7.10070670875801345571919352382, 7.37919937047148341501953087265, 7.79488126140241289617004408679, 7.87874864765669193479236167158, 8.893357507536551178093015173587, 9.106073597725074568894866773615, 9.541884241741345333064276369543, 9.571165917370971600287195208813
