| L(s) = 1 | − 3-s + 4·4-s + 3·5-s − 2·9-s − 4·12-s − 3·15-s + 12·16-s + 12·20-s − 18·23-s + 4·25-s + 5·27-s − 10·31-s − 8·36-s − 6·45-s + 24·47-s − 12·48-s − 14·49-s − 12·53-s − 12·60-s + 32·64-s + 18·69-s − 4·75-s + 36·80-s + 81-s − 72·92-s + 10·93-s + 16·100-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 2·4-s + 1.34·5-s − 2/3·9-s − 1.15·12-s − 0.774·15-s + 3·16-s + 2.68·20-s − 3.75·23-s + 4/5·25-s + 0.962·27-s − 1.79·31-s − 4/3·36-s − 0.894·45-s + 3.50·47-s − 1.73·48-s − 2·49-s − 1.64·53-s − 1.54·60-s + 4·64-s + 2.16·69-s − 0.461·75-s + 4.02·80-s + 1/9·81-s − 7.50·92-s + 1.03·93-s + 8/5·100-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.795471620\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.795471620\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T + p T^{2} \) |
| 5 | $C_2$ | \( 1 - 3 T + p T^{2} \) |
| 11 | $C_2$ | \( 1 + p T^{2} \) |
| good | 2 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) |
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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
−12.67036661082658082052048313460, −12.57931310526937732857742718164, −11.96696770341127170360323595282, −11.63447741611660777958485544718, −10.98894133422688914558602680253, −10.77062807594740952814869709491, −10.00425482064437413577053579651, −9.987392923331185778038405413004, −9.127927653632167176705081232588, −8.329573024704892966199754961875, −7.73025179398020348170243285461, −7.31904052608451463662153513888, −6.30832757896457576293815263842, −6.27427733884725445037706944222, −5.73816012184423975212059417801, −5.37630621914649892911840967897, −4.03353961381990640753076469189, −3.11665881992685121917386548775, −2.14740718118359859647804121198, −1.85776434273011545659431092329,
1.85776434273011545659431092329, 2.14740718118359859647804121198, 3.11665881992685121917386548775, 4.03353961381990640753076469189, 5.37630621914649892911840967897, 5.73816012184423975212059417801, 6.27427733884725445037706944222, 6.30832757896457576293815263842, 7.31904052608451463662153513888, 7.73025179398020348170243285461, 8.329573024704892966199754961875, 9.127927653632167176705081232588, 9.987392923331185778038405413004, 10.00425482064437413577053579651, 10.77062807594740952814869709491, 10.98894133422688914558602680253, 11.63447741611660777958485544718, 11.96696770341127170360323595282, 12.57931310526937732857742718164, 12.67036661082658082052048313460
