L(s) = 1 | − 2-s + 2·3-s − 4-s − 2·6-s − 7-s + 3·8-s + 3·9-s − 2·12-s + 14-s − 16-s − 3·18-s + 8·19-s − 2·21-s + 6·24-s − 6·25-s + 4·27-s + 28-s − 4·29-s − 5·32-s − 3·36-s + 12·37-s − 8·38-s + 2·42-s − 2·48-s + 49-s + 6·50-s + 12·53-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1.15·3-s − 1/2·4-s − 0.816·6-s − 0.377·7-s + 1.06·8-s + 9-s − 0.577·12-s + 0.267·14-s − 1/4·16-s − 0.707·18-s + 1.83·19-s − 0.436·21-s + 1.22·24-s − 6/5·25-s + 0.769·27-s + 0.188·28-s − 0.742·29-s − 0.883·32-s − 1/2·36-s + 1.97·37-s − 1.29·38-s + 0.308·42-s − 0.288·48-s + 1/7·49-s + 0.848·50-s + 1.64·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 49392 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49392 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.230683220\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.230683220\) |
\(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T + p T^{2} \) |
| 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | $C_1$ | \( 1 + T \) |
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 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
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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
−9.762171554631668381167652624967, −9.724661210726173973340366981146, −9.090946623633278801475280262159, −8.672365196683144779961098572501, −8.050951011396219930904197989085, −7.70869243915887411656672354688, −7.25047783802838427330028450541, −6.69032425884932976329953320783, −5.58120289551734980246344429859, −5.36367026412859974981569612163, −4.13559084050773741974089362056, −3.99570051430579360707309966688, −3.06814457319249350690123364058, −2.26612547708831611492323080497, −1.10536343915846851689160557608,
1.10536343915846851689160557608, 2.26612547708831611492323080497, 3.06814457319249350690123364058, 3.99570051430579360707309966688, 4.13559084050773741974089362056, 5.36367026412859974981569612163, 5.58120289551734980246344429859, 6.69032425884932976329953320783, 7.25047783802838427330028450541, 7.70869243915887411656672354688, 8.050951011396219930904197989085, 8.672365196683144779961098572501, 9.090946623633278801475280262159, 9.724661210726173973340366981146, 9.762171554631668381167652624967