L(s) = 1 | + 3-s − 7-s + 9-s − 6·13-s + 4·17-s − 4·19-s − 21-s − 23-s + 27-s − 8·29-s + 4·31-s − 2·37-s − 6·39-s − 4·43-s − 4·47-s + 49-s + 4·51-s + 8·53-s − 4·57-s + 10·61-s − 63-s − 4·67-s − 69-s − 12·71-s − 12·73-s + 4·79-s + 81-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.377·7-s + 1/3·9-s − 1.66·13-s + 0.970·17-s − 0.917·19-s − 0.218·21-s − 0.208·23-s + 0.192·27-s − 1.48·29-s + 0.718·31-s − 0.328·37-s − 0.960·39-s − 0.609·43-s − 0.583·47-s + 1/7·49-s + 0.560·51-s + 1.09·53-s − 0.529·57-s + 1.28·61-s − 0.125·63-s − 0.488·67-s − 0.120·69-s − 1.42·71-s − 1.40·73-s + 0.450·79-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 96600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 96600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.323355419\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.323355419\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 + T \) |
good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 8 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 12 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−13.80781567975664, −13.19693990670487, −12.93309906115734, −12.31655110640654, −11.94226826240492, −11.47568825860772, −10.67582656519655, −10.17000157667735, −9.855140433534136, −9.430469024413153, −8.761617683333625, −8.353516815143742, −7.671502038075133, −7.319100196672591, −6.862218811037455, −6.153554818435580, −5.576845390566581, −5.011131129141966, −4.423604410499007, −3.817726258632385, −3.240919234112200, −2.598598474895493, −2.117580593193453, −1.392176762588905, −0.3371289101746940,
0.3371289101746940, 1.392176762588905, 2.117580593193453, 2.598598474895493, 3.240919234112200, 3.817726258632385, 4.423604410499007, 5.011131129141966, 5.576845390566581, 6.153554818435580, 6.862218811037455, 7.319100196672591, 7.671502038075133, 8.353516815143742, 8.761617683333625, 9.430469024413153, 9.855140433534136, 10.17000157667735, 10.67582656519655, 11.47568825860772, 11.94226826240492, 12.31655110640654, 12.93309906115734, 13.19693990670487, 13.80781567975664