| L(s) = 1 | + 3-s + 3·7-s + 9-s − 11-s + 5·17-s − 8·19-s + 3·21-s + 27-s + 29-s + 3·31-s − 33-s − 8·37-s + 2·41-s + 8·43-s + 11·47-s + 2·49-s + 5·51-s + 11·53-s − 8·57-s + 5·59-s + 61-s + 3·63-s − 3·67-s + 16·71-s − 4·73-s − 3·77-s − 12·79-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.13·7-s + 1/3·9-s − 0.301·11-s + 1.21·17-s − 1.83·19-s + 0.654·21-s + 0.192·27-s + 0.185·29-s + 0.538·31-s − 0.174·33-s − 1.31·37-s + 0.312·41-s + 1.21·43-s + 1.60·47-s + 2/7·49-s + 0.700·51-s + 1.51·53-s − 1.05·57-s + 0.650·59-s + 0.128·61-s + 0.377·63-s − 0.366·67-s + 1.89·71-s − 0.468·73-s − 0.341·77-s − 1.35·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 202800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 202800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.338292491\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.338292491\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 13 | \( 1 \) |
| good | 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 11 T + p T^{2} \) |
| 53 | \( 1 - 11 T + p T^{2} \) |
| 59 | \( 1 - 5 T + p T^{2} \) |
| 61 | \( 1 - T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 - 3 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−12.95933729499657, −12.62663039120275, −12.15236260036461, −11.71305871206454, −11.10294336477363, −10.61018418039787, −10.32547732095867, −9.854848812387650, −9.100411853255212, −8.660766718471088, −8.381700710895569, −7.876664972776903, −7.380473135594449, −6.993995954086565, −6.246079445205715, −5.732946308523167, −5.216793708537312, −4.684860401750752, −4.080581386555389, −3.785664174371563, −2.937177026246080, −2.353246275264911, −1.971436860008562, −1.215994527310202, −0.5882925658851966,
0.5882925658851966, 1.215994527310202, 1.971436860008562, 2.353246275264911, 2.937177026246080, 3.785664174371563, 4.080581386555389, 4.684860401750752, 5.216793708537312, 5.732946308523167, 6.246079445205715, 6.993995954086565, 7.380473135594449, 7.876664972776903, 8.381700710895569, 8.660766718471088, 9.100411853255212, 9.854848812387650, 10.32547732095867, 10.61018418039787, 11.10294336477363, 11.71305871206454, 12.15236260036461, 12.62663039120275, 12.95933729499657
