| L(s) = 1 | + 3-s + 9-s − 4·13-s − 8·19-s + 25-s + 27-s − 20·37-s − 4·39-s − 8·43-s − 14·49-s − 8·57-s − 4·61-s − 24·67-s + 20·73-s + 75-s + 81-s + 4·97-s + 32·103-s + 28·109-s − 20·111-s − 4·117-s − 6·121-s + 127-s − 8·129-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1/3·9-s − 1.10·13-s − 1.83·19-s + 1/5·25-s + 0.192·27-s − 3.28·37-s − 0.640·39-s − 1.21·43-s − 2·49-s − 1.05·57-s − 0.512·61-s − 2.93·67-s + 2.34·73-s + 0.115·75-s + 1/9·81-s + 0.406·97-s + 3.15·103-s + 2.68·109-s − 1.89·111-s − 0.369·117-s − 0.545·121-s + 0.0887·127-s − 0.704·129-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 172800 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 172800 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_1$ | \( 1 - T \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
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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
−8.982199030019329801304860528684, −8.330811446905088069540074876671, −8.270140448885357284476919824336, −7.40264879408133961073539513214, −7.12360478295630625617848815878, −6.55157891837643531715592264808, −6.13587545605028508104380795249, −5.32469243442401148699352691061, −4.68831052899409581492283307982, −4.54153907278640853760279816056, −3.39274372810076015084631955767, −3.29035819756263414840801407793, −2.11935007719153584295308930473, −1.82346513010869405208007428430, 0,
1.82346513010869405208007428430, 2.11935007719153584295308930473, 3.29035819756263414840801407793, 3.39274372810076015084631955767, 4.54153907278640853760279816056, 4.68831052899409581492283307982, 5.32469243442401148699352691061, 6.13587545605028508104380795249, 6.55157891837643531715592264808, 7.12360478295630625617848815878, 7.40264879408133961073539513214, 8.270140448885357284476919824336, 8.330811446905088069540074876671, 8.982199030019329801304860528684
