| L(s) = 1 | − 2·3-s − 4·7-s − 3·9-s + 2·11-s − 4·16-s + 8·21-s − 9·25-s + 14·27-s − 4·33-s + 3·37-s − 16·41-s + 16·47-s + 8·48-s − 2·49-s − 12·53-s + 12·63-s − 14·67-s − 6·71-s + 8·73-s + 18·75-s − 8·77-s − 4·81-s − 12·83-s − 6·99-s + 4·101-s + 36·107-s − 6·111-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 1.51·7-s − 9-s + 0.603·11-s − 16-s + 1.74·21-s − 9/5·25-s + 2.69·27-s − 0.696·33-s + 0.493·37-s − 2.49·41-s + 2.33·47-s + 1.15·48-s − 2/7·49-s − 1.64·53-s + 1.51·63-s − 1.71·67-s − 0.712·71-s + 0.936·73-s + 2.07·75-s − 0.911·77-s − 4/9·81-s − 1.31·83-s − 0.603·99-s + 0.398·101-s + 3.48·107-s − 0.569·111-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 165649 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 165649 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2648290558\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2648290558\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 11 | $C_1$ | \( ( 1 - T )^{2} \) |
| 37 | $C_2$ | \( 1 - 3 T + p T^{2} \) |
| good | 2 | $C_2$ | \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 3 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
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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
−9.265168361059297487413940716234, −8.620233479065114660907179909222, −8.603539619290756001226038948684, −7.60269034011083985842455470421, −7.18568052203217455673362262430, −6.36261389471308870138602900888, −6.33639822785340552250290919527, −5.95243813251172409069674538549, −5.31834830440580903659775727248, −4.75334431768646572932470282063, −4.05438716306474070363726653663, −3.30474114837290393749034684472, −2.89084127074252720454161353058, −1.89867661786411457951020449167, −0.34282590174803099070000182310,
0.34282590174803099070000182310, 1.89867661786411457951020449167, 2.89084127074252720454161353058, 3.30474114837290393749034684472, 4.05438716306474070363726653663, 4.75334431768646572932470282063, 5.31834830440580903659775727248, 5.95243813251172409069674538549, 6.33639822785340552250290919527, 6.36261389471308870138602900888, 7.18568052203217455673362262430, 7.60269034011083985842455470421, 8.603539619290756001226038948684, 8.620233479065114660907179909222, 9.265168361059297487413940716234
