| L(s) = 1 | + 2-s − 4-s − 5-s + 7-s − 3·8-s − 10-s + 2·11-s − 2·13-s + 14-s − 16-s − 5·17-s + 2·19-s + 20-s + 2·22-s − 6·23-s − 4·25-s − 2·26-s − 28-s + 4·31-s + 5·32-s − 5·34-s − 35-s + 5·37-s + 2·38-s + 3·40-s + 9·41-s + 3·43-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s − 0.447·5-s + 0.377·7-s − 1.06·8-s − 0.316·10-s + 0.603·11-s − 0.554·13-s + 0.267·14-s − 1/4·16-s − 1.21·17-s + 0.458·19-s + 0.223·20-s + 0.426·22-s − 1.25·23-s − 4/5·25-s − 0.392·26-s − 0.188·28-s + 0.718·31-s + 0.883·32-s − 0.857·34-s − 0.169·35-s + 0.821·37-s + 0.324·38-s + 0.474·40-s + 1.40·41-s + 0.457·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 52983 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 52983 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.531017076\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.531017076\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 29 | \( 1 \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 5 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 - 3 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 + T + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.24785376357031, −14.00300567482762, −13.58191219157223, −12.91520350836278, −12.44594132918005, −11.92319389087325, −11.59451431397894, −11.05832632183773, −10.33966647502818, −9.637967068406593, −9.362016370648272, −8.705138576412308, −8.156803867196788, −7.688590449625243, −6.991701560414720, −6.382415685157006, −5.740068173928312, −5.371897777087672, −4.375432046033183, −4.244578107747571, −3.850915615823346, −2.760332092665474, −2.405674830820027, −1.335010139053379, −0.3988389920182563,
0.3988389920182563, 1.335010139053379, 2.405674830820027, 2.760332092665474, 3.850915615823346, 4.244578107747571, 4.375432046033183, 5.371897777087672, 5.740068173928312, 6.382415685157006, 6.991701560414720, 7.688590449625243, 8.156803867196788, 8.705138576412308, 9.362016370648272, 9.637967068406593, 10.33966647502818, 11.05832632183773, 11.59451431397894, 11.92319389087325, 12.44594132918005, 12.91520350836278, 13.58191219157223, 14.00300567482762, 14.24785376357031
