| L(s) = 1 | + 2-s + 4-s + 2·5-s + 7-s + 8-s − 3·9-s + 2·10-s + 2·11-s + 2·13-s + 14-s + 16-s − 3·18-s − 6·19-s + 2·20-s + 2·22-s + 23-s − 25-s + 2·26-s + 28-s − 4·31-s + 32-s + 2·35-s − 3·36-s + 4·37-s − 6·38-s + 2·40-s + 10·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.894·5-s + 0.377·7-s + 0.353·8-s − 9-s + 0.632·10-s + 0.603·11-s + 0.554·13-s + 0.267·14-s + 1/4·16-s − 0.707·18-s − 1.37·19-s + 0.447·20-s + 0.426·22-s + 0.208·23-s − 1/5·25-s + 0.392·26-s + 0.188·28-s − 0.718·31-s + 0.176·32-s + 0.338·35-s − 1/2·36-s + 0.657·37-s − 0.973·38-s + 0.316·40-s + 1.56·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 270802 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 270802 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.033462810\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.033462810\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 \) |
| good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 16 T + p T^{2} \) |
| 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
−12.87352913569824, −12.46529514839431, −11.84519287580940, −11.32309661380782, −11.09189114800853, −10.60172654117878, −10.17571333487547, −9.381538360798025, −9.107972530553972, −8.707067377722501, −8.052251749651472, −7.670053274834098, −6.962171225162799, −6.441776826203684, −6.042403969055955, −5.645339730976433, −5.352586168895558, −4.473932638779273, −4.140184708121501, −3.654899163908200, −2.879046978057572, −2.321150354077382, −2.057825792576287, −1.233054129651814, −0.5863155420100826,
0.5863155420100826, 1.233054129651814, 2.057825792576287, 2.321150354077382, 2.879046978057572, 3.654899163908200, 4.140184708121501, 4.473932638779273, 5.352586168895558, 5.645339730976433, 6.042403969055955, 6.441776826203684, 6.962171225162799, 7.670053274834098, 8.052251749651472, 8.707067377722501, 9.107972530553972, 9.381538360798025, 10.17571333487547, 10.60172654117878, 11.09189114800853, 11.32309661380782, 11.84519287580940, 12.46529514839431, 12.87352913569824
