L(s) = 1 | + 2·7-s − 3·9-s + 4·13-s − 16·19-s − 6·25-s − 16·31-s − 4·37-s + 8·43-s + 3·49-s − 12·61-s − 6·63-s + 8·67-s + 20·73-s − 32·79-s + 9·81-s + 8·91-s − 12·97-s + 32·103-s − 20·109-s − 12·117-s − 6·121-s + 127-s + 131-s − 32·133-s + 137-s + 139-s + 149-s + ⋯ |
L(s) = 1 | + 0.755·7-s − 9-s + 1.10·13-s − 3.67·19-s − 6/5·25-s − 2.87·31-s − 0.657·37-s + 1.21·43-s + 3/7·49-s − 1.53·61-s − 0.755·63-s + 0.977·67-s + 2.34·73-s − 3.60·79-s + 81-s + 0.838·91-s − 1.21·97-s + 3.15·103-s − 1.91·109-s − 1.10·117-s − 0.545·121-s + 0.0887·127-s + 0.0873·131-s − 2.77·133-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 112896 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112896 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + p T^{2} \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
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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
−9.117894034638208548514808203742, −8.560607806232820418165457845354, −8.471445676889714342866171635546, −7.930145034161771419271559855193, −7.27554251802063504690854958433, −6.67949194679031572918581545842, −6.02199704414356721368436371140, −5.82613210142107824134972732669, −5.19403339606859488056646370743, −4.18465932715971087267449515138, −4.12306698900858528561316116411, −3.27599538980291781109010644776, −2.14556791802447766709825303506, −1.86537117894268329726575064930, 0,
1.86537117894268329726575064930, 2.14556791802447766709825303506, 3.27599538980291781109010644776, 4.12306698900858528561316116411, 4.18465932715971087267449515138, 5.19403339606859488056646370743, 5.82613210142107824134972732669, 6.02199704414356721368436371140, 6.67949194679031572918581545842, 7.27554251802063504690854958433, 7.930145034161771419271559855193, 8.471445676889714342866171635546, 8.560607806232820418165457845354, 9.117894034638208548514808203742