| L(s) = 1 | + 2-s + 3-s − 5-s + 6-s − 7-s − 8-s − 10-s − 2·13-s − 14-s − 15-s − 16-s − 17-s − 2·19-s − 21-s + 23-s − 24-s + 25-s − 2·26-s − 27-s − 30-s − 34-s + 35-s − 2·38-s − 2·39-s + 40-s − 42-s + 4·43-s + ⋯ |
| L(s) = 1 | + 2-s + 3-s − 5-s + 6-s − 7-s − 8-s − 10-s − 2·13-s − 14-s − 15-s − 16-s − 17-s − 2·19-s − 21-s + 23-s − 24-s + 25-s − 2·26-s − 27-s − 30-s − 34-s + 35-s − 2·38-s − 2·39-s + 40-s − 42-s + 4·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3732624 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3732624 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8970724255\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8970724255\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2$ | \( 1 - T + T^{2} \) |
| 3 | $C_2$ | \( 1 - T + T^{2} \) |
| 7 | $C_2$ | \( 1 + T + T^{2} \) |
| 23 | $C_2$ | \( 1 - T + T^{2} \) |
| good | 5 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 17 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 31 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 43 | $C_1$ | \( ( 1 - T )^{4} \) |
| 47 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 53 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 67 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 73 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 89 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 97 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
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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
−9.805635274091665432834833788964, −9.023819932447814574509788148080, −9.010423892271369371727142574663, −8.334978778014852145802197846768, −8.161076805197861258857941592210, −7.55685547707300665244511654204, −7.21346306145627853832318704691, −6.85956693773364003787474625338, −6.30738990490504082904076190535, −6.19711346881874863819968477818, −5.35467900903824354420791166050, −5.05760324152608702594114047023, −4.44463851299407858716113124252, −4.34246508300763838210592622184, −3.68840924347350235531931147206, −3.49878571643793433078973281201, −2.74740420159459900000431919315, −2.36541981026713259333976269312, −2.33170297287602991500828736000, −0.51786949174969908421094164322,
0.51786949174969908421094164322, 2.33170297287602991500828736000, 2.36541981026713259333976269312, 2.74740420159459900000431919315, 3.49878571643793433078973281201, 3.68840924347350235531931147206, 4.34246508300763838210592622184, 4.44463851299407858716113124252, 5.05760324152608702594114047023, 5.35467900903824354420791166050, 6.19711346881874863819968477818, 6.30738990490504082904076190535, 6.85956693773364003787474625338, 7.21346306145627853832318704691, 7.55685547707300665244511654204, 8.161076805197861258857941592210, 8.334978778014852145802197846768, 9.010423892271369371727142574663, 9.023819932447814574509788148080, 9.805635274091665432834833788964
