| L(s) = 1 | + 2·7-s − 4·11-s − 4·13-s − 8·17-s + 4·23-s + 2·25-s + 4·29-s + 8·31-s + 4·37-s − 16·41-s − 16·43-s + 8·47-s + 3·49-s − 4·53-s + 16·59-s + 4·61-s + 8·67-s + 12·71-s + 12·73-s − 8·77-s + 8·79-s + 8·83-s − 8·91-s − 4·97-s − 24·101-s + 8·103-s − 4·107-s + ⋯ |
| L(s) = 1 | + 0.755·7-s − 1.20·11-s − 1.10·13-s − 1.94·17-s + 0.834·23-s + 2/5·25-s + 0.742·29-s + 1.43·31-s + 0.657·37-s − 2.49·41-s − 2.43·43-s + 1.16·47-s + 3/7·49-s − 0.549·53-s + 2.08·59-s + 0.512·61-s + 0.977·67-s + 1.42·71-s + 1.40·73-s − 0.911·77-s + 0.900·79-s + 0.878·83-s − 0.838·91-s − 0.406·97-s − 2.38·101-s + 0.788·103-s − 0.386·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16257024 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16257024 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.560796406\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.560796406\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 4 T + 14 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $D_{4}$ | \( 1 + 8 T + 38 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 4 T + 38 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 4 T + 14 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 8 T + 30 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $D_{4}$ | \( 1 + 16 T + 134 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 47 | $D_{4}$ | \( 1 - 8 T + 62 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 59 | $D_{4}$ | \( 1 - 16 T + 134 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 4 T + 78 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 8 T + 102 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 12 T + 166 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 12 T + 134 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 8 T + 126 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 166 T^{2} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 4 T + 150 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
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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
−8.509672252713730275777334532126, −8.444731779920151760281743928077, −7.82959200290526836460252068827, −7.71787817246087997202866334029, −7.01047081676214741190450824485, −6.81623333407414810570331582483, −6.52196166269636410034774062778, −6.27419421997996820185455579567, −5.30770811522963424351480483843, −5.05934167865780128187916004134, −5.04030801807626924344360398248, −4.75475663012181967374798556675, −3.97399529584438805659431538151, −3.80925427426975247268674329409, −2.90264462193344413982642871729, −2.75783847575321147976885301170, −2.17528332126608638850565245047, −1.96121602390600694359033349434, −1.05501993262064266688989517679, −0.38879986028860335242397617223,
0.38879986028860335242397617223, 1.05501993262064266688989517679, 1.96121602390600694359033349434, 2.17528332126608638850565245047, 2.75783847575321147976885301170, 2.90264462193344413982642871729, 3.80925427426975247268674329409, 3.97399529584438805659431538151, 4.75475663012181967374798556675, 5.04030801807626924344360398248, 5.05934167865780128187916004134, 5.30770811522963424351480483843, 6.27419421997996820185455579567, 6.52196166269636410034774062778, 6.81623333407414810570331582483, 7.01047081676214741190450824485, 7.71787817246087997202866334029, 7.82959200290526836460252068827, 8.444731779920151760281743928077, 8.509672252713730275777334532126
