| L(s) = 1 | + 2-s − 3-s + 4-s − 6-s + 8-s + 9-s − 12-s + 4·13-s + 16-s + 18-s + 12·23-s − 24-s − 10·25-s + 4·26-s − 27-s + 32-s + 36-s + 16·37-s − 4·39-s + 12·46-s − 24·47-s − 48-s − 10·49-s − 10·50-s + 4·52-s − 54-s + 24·59-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.408·6-s + 0.353·8-s + 1/3·9-s − 0.288·12-s + 1.10·13-s + 1/4·16-s + 0.235·18-s + 2.50·23-s − 0.204·24-s − 2·25-s + 0.784·26-s − 0.192·27-s + 0.176·32-s + 1/6·36-s + 2.63·37-s − 0.640·39-s + 1.76·46-s − 3.50·47-s − 0.144·48-s − 1.42·49-s − 1.41·50-s + 0.554·52-s − 0.136·54-s + 3.12·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 249696 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 249696 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.469840961\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.469840961\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 - T \) |
| 3 | $C_1$ | \( 1 + T \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
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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
−8.745924693621362183100177299272, −8.584868551115419667042947592500, −7.81014283472581039979750102513, −7.57323216500839527256914149425, −6.70932306148654350305246926067, −6.62762171992257665130856519803, −5.95967651264422152670977788910, −5.60715979666329449780909401068, −4.94376122657450226002634205632, −4.60759445774375135200288872743, −3.84609690192901140186989283204, −3.42809860399723990451499242436, −2.71372337045158087575655535987, −1.81193945201237649695550582777, −0.943136411131521382953657369703,
0.943136411131521382953657369703, 1.81193945201237649695550582777, 2.71372337045158087575655535987, 3.42809860399723990451499242436, 3.84609690192901140186989283204, 4.60759445774375135200288872743, 4.94376122657450226002634205632, 5.60715979666329449780909401068, 5.95967651264422152670977788910, 6.62762171992257665130856519803, 6.70932306148654350305246926067, 7.57323216500839527256914149425, 7.81014283472581039979750102513, 8.584868551115419667042947592500, 8.745924693621362183100177299272
