| L(s) = 1 | + 2·2-s − 4-s − 8·8-s + 6·9-s − 12·13-s − 7·16-s − 6·17-s + 12·18-s − 8·19-s + 2·25-s − 24·26-s + 14·32-s − 12·34-s − 6·36-s − 16·38-s + 2·43-s + 14·49-s + 4·50-s + 12·52-s − 20·53-s − 8·59-s + 35·64-s + 24·67-s + 6·68-s − 48·72-s + 8·76-s + 27·81-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 1/2·4-s − 2.82·8-s + 2·9-s − 3.32·13-s − 7/4·16-s − 1.45·17-s + 2.82·18-s − 1.83·19-s + 2/5·25-s − 4.70·26-s + 2.47·32-s − 2.05·34-s − 36-s − 2.59·38-s + 0.304·43-s + 2·49-s + 0.565·50-s + 1.66·52-s − 2.74·53-s − 1.04·59-s + 35/8·64-s + 2.93·67-s + 0.727·68-s − 5.65·72-s + 0.917·76-s + 3·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 534361 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 534361 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.259364731\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.259364731\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 17 | $C_2$ | \( 1 + 6 T + p T^{2} \) |
| 43 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 2 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 3 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 54 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 86 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 66 T^{2} + p^{2} T^{4} \) |
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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
−10.59583001653114290938375217107, −9.920980218199175777247727116978, −9.876534270704161658079177549607, −9.315394382465447076031384619868, −9.113835193659054859569412812982, −8.544691491285748005240295735382, −7.896257133584619477284324166440, −7.49037042412765520776622526021, −6.97368566273592317035358891543, −6.50272886005381834669674356092, −6.26015945516951342150663937168, −5.30009396925754397741252139028, −4.95122034908550438464846660952, −4.50885446851140731296298800602, −4.50721832258566562738863463406, −3.95940264624050294216987058470, −3.25113182902751259452617687386, −2.26222079344843177353092771431, −2.24182646941025045654032165856, −0.44729243187594714528199189926,
0.44729243187594714528199189926, 2.24182646941025045654032165856, 2.26222079344843177353092771431, 3.25113182902751259452617687386, 3.95940264624050294216987058470, 4.50721832258566562738863463406, 4.50885446851140731296298800602, 4.95122034908550438464846660952, 5.30009396925754397741252139028, 6.26015945516951342150663937168, 6.50272886005381834669674356092, 6.97368566273592317035358891543, 7.49037042412765520776622526021, 7.896257133584619477284324166440, 8.544691491285748005240295735382, 9.113835193659054859569412812982, 9.315394382465447076031384619868, 9.876534270704161658079177549607, 9.920980218199175777247727116978, 10.59583001653114290938375217107
