| L(s) = 1 | − 2·2-s + 2·4-s + 2·5-s − 4·10-s + 4·11-s + 6·13-s − 4·16-s − 6·17-s + 4·20-s − 8·22-s + 12·23-s − 7·25-s − 12·26-s + 8·32-s + 12·34-s − 6·37-s + 8·44-s − 24·46-s + 5·49-s + 14·50-s + 12·52-s + 8·55-s + 20·59-s − 8·64-s + 12·65-s + 24·67-s − 12·68-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 4-s + 0.894·5-s − 1.26·10-s + 1.20·11-s + 1.66·13-s − 16-s − 1.45·17-s + 0.894·20-s − 1.70·22-s + 2.50·23-s − 7/5·25-s − 2.35·26-s + 1.41·32-s + 2.05·34-s − 0.986·37-s + 1.20·44-s − 3.53·46-s + 5/7·49-s + 1.97·50-s + 1.66·52-s + 1.07·55-s + 2.60·59-s − 64-s + 1.48·65-s + 2.93·67-s − 1.45·68-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 876096 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 876096 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.391712796\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.391712796\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + p T + p T^{2} \) |
| 3 | | \( 1 \) |
| 13 | $C_2$ | \( 1 - 6 T + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 45 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 117 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 162 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
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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
−9.992638986484286305177440986783, −9.896031036831754310614677895701, −9.318769480271139168092393886138, −8.963966574754652868723293127536, −8.720224063261848675880045306886, −8.489778513412188484963770881819, −7.953424199830714189751546535025, −7.21038028289056137379673801635, −6.82376974626189334107760792119, −6.69002984096417571674870515759, −6.21658840685772715266108046972, −5.39988461629087636259830217889, −5.37050853715856357629698359101, −4.28217468648720941394710763754, −4.07455294406761503892822765197, −3.41304881685524990367262677816, −2.56668023260271273998744267207, −1.95342585048035458596849589691, −1.41932102265810360053244028427, −0.77671772229382756802961712828,
0.77671772229382756802961712828, 1.41932102265810360053244028427, 1.95342585048035458596849589691, 2.56668023260271273998744267207, 3.41304881685524990367262677816, 4.07455294406761503892822765197, 4.28217468648720941394710763754, 5.37050853715856357629698359101, 5.39988461629087636259830217889, 6.21658840685772715266108046972, 6.69002984096417571674870515759, 6.82376974626189334107760792119, 7.21038028289056137379673801635, 7.953424199830714189751546535025, 8.489778513412188484963770881819, 8.720224063261848675880045306886, 8.963966574754652868723293127536, 9.318769480271139168092393886138, 9.896031036831754310614677895701, 9.992638986484286305177440986783
