| L(s) = 1 | − 3-s − 3·7-s + 3·21-s − 3·23-s + 27-s − 29-s − 41-s + 3·47-s + 5·49-s − 61-s − 3·67-s + 3·69-s − 81-s + 3·83-s + 87-s + 2·89-s − 2·101-s − 2·109-s − 121-s + 123-s + 127-s + 131-s + 137-s + 139-s − 3·141-s − 5·147-s + 149-s + ⋯ |
| L(s) = 1 | − 3-s − 3·7-s + 3·21-s − 3·23-s + 27-s − 29-s − 41-s + 3·47-s + 5·49-s − 61-s − 3·67-s + 3·69-s − 81-s + 3·83-s + 87-s + 2·89-s − 2·101-s − 2·109-s − 121-s + 123-s + 127-s + 131-s + 137-s + 139-s − 3·141-s − 5·147-s + 149-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12960000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12960000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.07437147082\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.07437147082\) |
| \(L(1)\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( 1 + T + T^{2} \) |
| 5 | | \( 1 \) |
| good | 7 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 + T + T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 17 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 23 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 + T + T^{2} ) \) |
| 29 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 41 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 47 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 - T + T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 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_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 73 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 83 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 - T + T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
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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
−8.950471305527200165913420314449, −8.830543883062758526579946750916, −7.950821820276303306155377821785, −7.84801008363821895872030989119, −7.35014736276083701131312028986, −6.80518579417157308514338345250, −6.69321995569481288978389517780, −6.09038764979851478767533676633, −6.04059374890192725873992573803, −5.80180359799048099296536788634, −5.40602217646731782991194303894, −4.73977578504509466980451232513, −4.16922672851381335766982729892, −3.89103169893303500081729982636, −3.43693181128580221651685283601, −3.16050658357450261029861473460, −2.41442422883984642557358974120, −2.24532460033465538263627394744, −1.20214410967519935251523772018, −0.17670312809240730142134776671,
0.17670312809240730142134776671, 1.20214410967519935251523772018, 2.24532460033465538263627394744, 2.41442422883984642557358974120, 3.16050658357450261029861473460, 3.43693181128580221651685283601, 3.89103169893303500081729982636, 4.16922672851381335766982729892, 4.73977578504509466980451232513, 5.40602217646731782991194303894, 5.80180359799048099296536788634, 6.04059374890192725873992573803, 6.09038764979851478767533676633, 6.69321995569481288978389517780, 6.80518579417157308514338345250, 7.35014736276083701131312028986, 7.84801008363821895872030989119, 7.950821820276303306155377821785, 8.830543883062758526579946750916, 8.950471305527200165913420314449
