| L(s) = 1 | − 3·3-s − 4·4-s − 9·7-s + 6·9-s + 12·12-s − 7·13-s + 12·16-s − 6·19-s + 27·21-s + 5·25-s − 9·27-s + 36·28-s − 7·31-s − 24·36-s − 9·37-s + 21·39-s − 8·43-s − 36·48-s + 47·49-s + 28·52-s + 18·57-s − 12·61-s − 54·63-s − 32·64-s + 5·67-s − 27·73-s − 15·75-s + ⋯ |
| L(s) = 1 | − 1.73·3-s − 2·4-s − 3.40·7-s + 2·9-s + 3.46·12-s − 1.94·13-s + 3·16-s − 1.37·19-s + 5.89·21-s + 25-s − 1.73·27-s + 6.80·28-s − 1.25·31-s − 4·36-s − 1.47·37-s + 3.36·39-s − 1.21·43-s − 5.19·48-s + 47/7·49-s + 3.88·52-s + 2.38·57-s − 1.53·61-s − 6.80·63-s − 4·64-s + 0.610·67-s − 3.16·73-s − 1.73·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16641 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16641 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + p T + p T^{2} \) |
| 43 | $C_2$ | \( 1 + 8 T + p T^{2} \) |
| good | 2 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 5 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 10 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
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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
−12.80543620150534401636915604552, −12.77059704863911707217640644848, −12.23614595780406933556606832717, −12.00321542250815881352977853256, −10.55490355283231007832515842370, −10.47486865891372303677502943847, −9.806502953952050504656481984844, −9.726773278695380431056419435563, −9.065248595650768776568045985041, −8.633915169686977569657784958178, −7.18655804081985587007586576971, −7.10683630188130986197507329790, −6.19386230426811351510762397112, −5.90043185219449490102402378679, −5.08935041145173611379631339493, −4.59892215297977124356818671545, −3.76827330031665716512726383843, −3.09116692298886609499007591441, 0, 0,
3.09116692298886609499007591441, 3.76827330031665716512726383843, 4.59892215297977124356818671545, 5.08935041145173611379631339493, 5.90043185219449490102402378679, 6.19386230426811351510762397112, 7.10683630188130986197507329790, 7.18655804081985587007586576971, 8.633915169686977569657784958178, 9.065248595650768776568045985041, 9.726773278695380431056419435563, 9.806502953952050504656481984844, 10.47486865891372303677502943847, 10.55490355283231007832515842370, 12.00321542250815881352977853256, 12.23614595780406933556606832717, 12.77059704863911707217640644848, 12.80543620150534401636915604552
