| L(s) = 1 | + 2-s − 4-s − 3·8-s − 16-s + 10·25-s + 4·29-s + 5·32-s + 12·37-s − 7·49-s + 10·50-s + 20·53-s + 4·58-s + 7·64-s + 12·74-s − 7·98-s − 10·100-s + 20·106-s − 36·109-s − 4·113-s − 4·116-s − 6·121-s + 127-s − 3·128-s + 131-s + 137-s + 139-s − 12·148-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s − 1.06·8-s − 1/4·16-s + 2·25-s + 0.742·29-s + 0.883·32-s + 1.97·37-s − 49-s + 1.41·50-s + 2.74·53-s + 0.525·58-s + 7/8·64-s + 1.39·74-s − 0.707·98-s − 100-s + 1.94·106-s − 3.44·109-s − 0.376·113-s − 0.371·116-s − 0.545·121-s + 0.0887·127-s − 0.265·128-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 0.986·148-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 63504 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63504 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.648168200\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.648168200\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + p T^{2} \) |
| 3 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
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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
−12.48581193760422415313143247331, −11.87554326925152230221395520467, −11.44873823039463598382753223712, −10.96014345974379194782959146475, −10.27664416446270182937686146892, −10.03384630134626803101284451735, −9.247886603319706769364924249349, −8.990776813701639835110735773678, −8.447308015944748974716475388576, −7.944170599950724135117929823773, −7.30327994275901525197194475622, −6.45656284749156489693295988644, −6.41497312201832572772558133114, −5.30825754932605045507871126943, −5.23447044052488790323012056384, −4.32704436396164722006082050468, −4.01509482432327481739222024615, −3.01526999505069296776811776982, −2.58754309290365295518166661006, −0.998177539737668739040491763176,
0.998177539737668739040491763176, 2.58754309290365295518166661006, 3.01526999505069296776811776982, 4.01509482432327481739222024615, 4.32704436396164722006082050468, 5.23447044052488790323012056384, 5.30825754932605045507871126943, 6.41497312201832572772558133114, 6.45656284749156489693295988644, 7.30327994275901525197194475622, 7.944170599950724135117929823773, 8.447308015944748974716475388576, 8.990776813701639835110735773678, 9.247886603319706769364924249349, 10.03384630134626803101284451735, 10.27664416446270182937686146892, 10.96014345974379194782959146475, 11.44873823039463598382753223712, 11.87554326925152230221395520467, 12.48581193760422415313143247331
