L(s) = 1 | + 2·2-s + 3·4-s + 7-s + 4·8-s − 5·9-s − 6·11-s + 2·14-s + 5·16-s − 10·18-s − 12·22-s − 6·23-s − 10·25-s + 3·28-s + 6·32-s − 15·36-s − 14·37-s + 16·43-s − 18·44-s − 12·46-s + 49-s − 20·50-s − 24·53-s + 4·56-s − 5·63-s + 7·64-s + 10·67-s + 24·71-s + ⋯ |
L(s) = 1 | + 1.41·2-s + 3/2·4-s + 0.377·7-s + 1.41·8-s − 5/3·9-s − 1.80·11-s + 0.534·14-s + 5/4·16-s − 2.35·18-s − 2.55·22-s − 1.25·23-s − 2·25-s + 0.566·28-s + 1.06·32-s − 5/2·36-s − 2.30·37-s + 2.43·43-s − 2.71·44-s − 1.76·46-s + 1/7·49-s − 2.82·50-s − 3.29·53-s + 0.534·56-s − 0.629·63-s + 7/8·64-s + 1.22·67-s + 2.84·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 231868 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 231868 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | $C_1$ | \( 1 - T \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
good | 3 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
| 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
−8.357542398475312637001956914796, −8.284684574830567460945123276136, −7.63380170955425435565378568582, −7.56044433828328293164495992231, −6.55543304367446106392236047763, −6.10743185313712551583625613621, −5.72895855893330977484350370148, −5.19256942155997860504028721022, −5.06509234546123458101387314677, −4.16724531605991793017887414807, −3.60858467418748472558474969025, −3.06437335560668526609552925191, −2.33000973337707432224237511883, −2.01051836387845799731692253165, 0,
2.01051836387845799731692253165, 2.33000973337707432224237511883, 3.06437335560668526609552925191, 3.60858467418748472558474969025, 4.16724531605991793017887414807, 5.06509234546123458101387314677, 5.19256942155997860504028721022, 5.72895855893330977484350370148, 6.10743185313712551583625613621, 6.55543304367446106392236047763, 7.56044433828328293164495992231, 7.63380170955425435565378568582, 8.284684574830567460945123276136, 8.357542398475312637001956914796