| L(s) = 1 | + 2·3-s − 2·4-s − 2·5-s − 15·9-s + 14·11-s − 4·12-s − 4·15-s + 4·16-s + 4·20-s + 34·23-s − 47·25-s − 50·27-s + 34·31-s + 28·33-s + 30·36-s + 94·37-s − 28·44-s + 30·45-s − 116·47-s + 8·48-s + 26·49-s + 4·53-s − 28·55-s − 110·59-s + 8·60-s − 8·64-s + 178·67-s + ⋯ |
| L(s) = 1 | + 2/3·3-s − 1/2·4-s − 2/5·5-s − 5/3·9-s + 1.27·11-s − 1/3·12-s − 0.266·15-s + 1/4·16-s + 1/5·20-s + 1.47·23-s − 1.87·25-s − 1.85·27-s + 1.09·31-s + 0.848·33-s + 5/6·36-s + 2.54·37-s − 0.636·44-s + 2/3·45-s − 2.46·47-s + 1/6·48-s + 0.530·49-s + 4/53·53-s − 0.509·55-s − 1.86·59-s + 2/15·60-s − 1/8·64-s + 2.65·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 484 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 484 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.8001781479\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8001781479\) |
| \(L(2)\) |
|
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 | $C_2$ | \( 1 + p T^{2} \) |
| 11 | $C_2$ | \( 1 - 14 T + p^{2} T^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - T + p^{2} T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 + T + p^{2} T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 - 26 T^{2} + p^{4} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 266 T^{2} + p^{4} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 70 T^{2} + p^{4} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 74 T^{2} + p^{4} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 17 T + p^{2} T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 530 T^{2} + p^{4} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 17 T + p^{2} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 47 T + p^{2} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 3290 T^{2} + p^{4} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 3410 T^{2} + p^{4} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 58 T + p^{2} T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 2 T + p^{2} T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 55 T + p^{2} T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 - 242 T^{2} + p^{4} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 89 T + p^{2} T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 7 T + p^{2} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 5542 T^{2} + p^{4} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 11330 T^{2} + p^{4} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 12626 T^{2} + p^{4} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 97 T + p^{2} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 121 T + p^{2} T^{2} )^{2} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−18.03853100477620250796432743122, −17.40262567579664285546855982053, −16.98888891540953619011722420661, −16.46854806215560065084358066920, −15.30839630622531230607954830551, −14.94091070994451609486410455958, −14.29477702480245575692025488584, −13.82938194981975101368914151988, −13.19338430644637928044773230956, −12.21185689093297284948418677747, −11.31560845057126233909142094495, −11.30230148601888066512357865305, −9.559499288456436870377875571410, −9.401252051667500770697801645388, −8.295168302387767806850181246400, −8.025953340221647977566280143571, −6.58172764741867550155760262949, −5.62852715127506949887252766425, −4.22152064120141929741538303524, −3.03710504950651394798101376906,
3.03710504950651394798101376906, 4.22152064120141929741538303524, 5.62852715127506949887252766425, 6.58172764741867550155760262949, 8.025953340221647977566280143571, 8.295168302387767806850181246400, 9.401252051667500770697801645388, 9.559499288456436870377875571410, 11.30230148601888066512357865305, 11.31560845057126233909142094495, 12.21185689093297284948418677747, 13.19338430644637928044773230956, 13.82938194981975101368914151988, 14.29477702480245575692025488584, 14.94091070994451609486410455958, 15.30839630622531230607954830551, 16.46854806215560065084358066920, 16.98888891540953619011722420661, 17.40262567579664285546855982053, 18.03853100477620250796432743122
