| L(s) = 1 | + 9·3-s + 8·4-s − 34·7-s + 54·9-s + 72·12-s + 159·13-s − 56·19-s − 306·21-s − 125·25-s + 243·27-s − 272·28-s + 432·36-s + 1.43e3·39-s − 71·43-s + 181·49-s + 1.27e3·52-s − 504·57-s + 719·61-s − 1.83e3·63-s − 512·64-s − 753·67-s + 271·73-s − 1.12e3·75-s − 448·76-s − 381·79-s + 729·81-s − 2.44e3·84-s + ⋯ |
| L(s) = 1 | + 1.73·3-s + 4-s − 1.83·7-s + 2·9-s + 1.73·12-s + 3.39·13-s − 0.676·19-s − 3.17·21-s − 25-s + 1.73·27-s − 1.83·28-s + 2·36-s + 5.87·39-s − 0.251·43-s + 0.527·49-s + 3.39·52-s − 1.17·57-s + 1.50·61-s − 3.67·63-s − 64-s − 1.37·67-s + 0.434·73-s − 1.73·75-s − 0.676·76-s − 0.542·79-s + 81-s − 3.17·84-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3249 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3249 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.480913423\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.480913423\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( 1 - p^{2} T + p^{3} T^{2} \) |
| 19 | $C_2$ | \( 1 + 56 T + p^{3} T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - p^{3} T^{2} + p^{6} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + p^{3} T^{2} + p^{6} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 17 T + p^{3} T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 89 T + p^{3} T^{2} )( 1 - 70 T + p^{3} T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + p^{3} T^{2} + p^{6} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + p^{3} T^{2} + p^{6} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - p^{3} T^{2} + p^{6} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 19 T + p^{3} T^{2} )( 1 + 19 T + p^{3} T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 323 T + p^{3} T^{2} )( 1 + 323 T + p^{3} T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - p^{3} T^{2} + p^{6} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 449 T + p^{3} T^{2} )( 1 + 520 T + p^{3} T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + p^{3} T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - p^{3} T^{2} + p^{6} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - p^{3} T^{2} + p^{6} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 901 T + p^{3} T^{2} )( 1 + 182 T + p^{3} T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 127 T + p^{3} T^{2} )( 1 + 880 T + p^{3} T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - p^{3} T^{2} + p^{6} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 1190 T + p^{3} T^{2} )( 1 + 919 T + p^{3} T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 503 T + p^{3} T^{2} )( 1 + 884 T + p^{3} T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - p^{3} T^{2} + p^{6} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 523 T + p^{3} T^{2} )( 1 + 1853 T + p^{3} T^{2} ) \) |
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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
−15.24562440126361820947066311545, −14.39567743145523192132462223026, −13.63608625370075606710238874355, −13.43944730235544544541079656162, −13.03302569385269020941344652807, −12.44483600085620753747819404727, −11.38498782564347834791691070442, −11.02032282926628718680724595408, −10.18185654626571010884123998587, −9.724201322626407553451096896728, −8.806639002151157856738958306496, −8.688216143918410257488473023725, −7.86540357229953949692796742019, −6.98152877684143104668063435028, −6.32163775233853013000295558545, −6.06117035102681201774814687031, −3.94637740306862740793326727599, −3.56439660893834695173895439541, −2.80406245202067847754825271536, −1.61753378596895613340618060840,
1.61753378596895613340618060840, 2.80406245202067847754825271536, 3.56439660893834695173895439541, 3.94637740306862740793326727599, 6.06117035102681201774814687031, 6.32163775233853013000295558545, 6.98152877684143104668063435028, 7.86540357229953949692796742019, 8.688216143918410257488473023725, 8.806639002151157856738958306496, 9.724201322626407553451096896728, 10.18185654626571010884123998587, 11.02032282926628718680724595408, 11.38498782564347834791691070442, 12.44483600085620753747819404727, 13.03302569385269020941344652807, 13.43944730235544544541079656162, 13.63608625370075606710238874355, 14.39567743145523192132462223026, 15.24562440126361820947066311545
