| L(s) = 1 | + 3·3-s − 2·5-s + 6·9-s − 6·15-s − 4·16-s − 6·17-s + 3·25-s + 9·27-s + 12·41-s − 12·43-s − 12·45-s − 18·47-s − 12·48-s − 18·51-s − 12·59-s − 28·67-s + 9·75-s − 2·79-s + 8·80-s + 9·81-s + 24·83-s + 12·85-s + 24·89-s − 36·101-s − 30·109-s − 21·121-s + 36·123-s + ⋯ |
| L(s) = 1 | + 1.73·3-s − 0.894·5-s + 2·9-s − 1.54·15-s − 16-s − 1.45·17-s + 3/5·25-s + 1.73·27-s + 1.87·41-s − 1.82·43-s − 1.78·45-s − 2.62·47-s − 1.73·48-s − 2.52·51-s − 1.56·59-s − 3.42·67-s + 1.03·75-s − 0.225·79-s + 0.894·80-s + 81-s + 2.63·83-s + 1.30·85-s + 2.54·89-s − 3.58·101-s − 2.87·109-s − 1.90·121-s + 3.24·123-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 540225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 540225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | $C_2$ | \( 1 - p T + p T^{2} \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | | \( 1 \) |
| good | 2 | $C_2$ | \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
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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
−8.210184381994137443519230420468, −7.79509008882088804636414623325, −7.67498052105787514083614275616, −6.86443729079700164782083564306, −6.67134380816033742861497166759, −6.22556914735544177770859299371, −5.21052441685335331437171683372, −4.64183738354361936328487143401, −4.33023383010768085745155570778, −3.90569825188449594842439463254, −3.07565995342284756040604408072, −2.95540115752479584920936053969, −2.09497145721286844219096031032, −1.56775892122303114558742737725, 0,
1.56775892122303114558742737725, 2.09497145721286844219096031032, 2.95540115752479584920936053969, 3.07565995342284756040604408072, 3.90569825188449594842439463254, 4.33023383010768085745155570778, 4.64183738354361936328487143401, 5.21052441685335331437171683372, 6.22556914735544177770859299371, 6.67134380816033742861497166759, 6.86443729079700164782083564306, 7.67498052105787514083614275616, 7.79509008882088804636414623325, 8.210184381994137443519230420468
