| L(s) = 1 | + 3-s + 4·5-s − 7-s + 9-s + 4·15-s + 12·17-s − 21-s + 2·25-s + 27-s − 4·35-s − 20·37-s − 20·41-s + 24·43-s + 4·45-s − 16·47-s + 49-s + 12·51-s + 8·59-s − 63-s + 24·67-s + 2·75-s + 16·79-s + 81-s + 8·83-s + 48·85-s + 12·89-s + 20·101-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.78·5-s − 0.377·7-s + 1/3·9-s + 1.03·15-s + 2.91·17-s − 0.218·21-s + 2/5·25-s + 0.192·27-s − 0.676·35-s − 3.28·37-s − 3.12·41-s + 3.65·43-s + 0.596·45-s − 2.33·47-s + 1/7·49-s + 1.68·51-s + 1.04·59-s − 0.125·63-s + 2.93·67-s + 0.230·75-s + 1.80·79-s + 1/9·81-s + 0.878·83-s + 5.20·85-s + 1.27·89-s + 1.99·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 592704 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 592704 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.676818641\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.676818641\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 - T \) |
| 7 | $C_1$ | \( 1 + T \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
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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
−8.577574048820412073979821096434, −7.86004275519657164377594722946, −7.65994621019403063529610936786, −7.05267897803070418923661148552, −6.42136658293515809263418102895, −6.27830454481138490924310987858, −5.47919414246685677243404259713, −5.25019213638990204722565722077, −5.06048784569425782183212035239, −3.72769041359201298820498364558, −3.56921408081238054859871707771, −3.08157339545259112628074418683, −2.06515692945354055956624655608, −1.91473513602758200866610556345, −1.00371826659586316798708144945,
1.00371826659586316798708144945, 1.91473513602758200866610556345, 2.06515692945354055956624655608, 3.08157339545259112628074418683, 3.56921408081238054859871707771, 3.72769041359201298820498364558, 5.06048784569425782183212035239, 5.25019213638990204722565722077, 5.47919414246685677243404259713, 6.27830454481138490924310987858, 6.42136658293515809263418102895, 7.05267897803070418923661148552, 7.65994621019403063529610936786, 7.86004275519657164377594722946, 8.577574048820412073979821096434
