| L(s) = 1 | − 3-s + 9-s − 2·11-s − 3·13-s − 5·17-s + 4·19-s − 5·23-s − 27-s + 9·29-s − 5·31-s + 2·33-s − 10·37-s + 3·39-s − 41-s + 3·43-s − 12·47-s + 5·51-s − 11·53-s − 4·57-s + 5·59-s − 9·61-s − 4·67-s + 5·69-s + 4·71-s − 6·73-s − 14·79-s + 81-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/3·9-s − 0.603·11-s − 0.832·13-s − 1.21·17-s + 0.917·19-s − 1.04·23-s − 0.192·27-s + 1.67·29-s − 0.898·31-s + 0.348·33-s − 1.64·37-s + 0.480·39-s − 0.156·41-s + 0.457·43-s − 1.75·47-s + 0.700·51-s − 1.51·53-s − 0.529·57-s + 0.650·59-s − 1.15·61-s − 0.488·67-s + 0.601·69-s + 0.474·71-s − 0.702·73-s − 1.57·79-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 58800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 58800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2760314881\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2760314881\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| good | 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 5 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + T + p T^{2} \) |
| 43 | \( 1 - 3 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 11 T + p T^{2} \) |
| 59 | \( 1 - 5 T + p T^{2} \) |
| 61 | \( 1 + 9 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 + 5 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−14.22301141913599, −13.88316733584741, −13.30224151797391, −12.69584662129272, −12.33059447229120, −11.80531209488317, −11.30821314072776, −10.80535235426802, −10.17205686611789, −9.903506245618586, −9.270831632098775, −8.559434587519221, −8.163917540522951, −7.321281457573941, −7.139985760005285, −6.331920959921979, −5.949082263087836, −5.061332056436096, −4.888924159785384, −4.229242230649509, −3.382774714235262, −2.789614659421227, −2.019116079175144, −1.384745889462398, −0.1830062427902622,
0.1830062427902622, 1.384745889462398, 2.019116079175144, 2.789614659421227, 3.382774714235262, 4.229242230649509, 4.888924159785384, 5.061332056436096, 5.949082263087836, 6.331920959921979, 7.139985760005285, 7.321281457573941, 8.163917540522951, 8.559434587519221, 9.270831632098775, 9.903506245618586, 10.17205686611789, 10.80535235426802, 11.30821314072776, 11.80531209488317, 12.33059447229120, 12.69584662129272, 13.30224151797391, 13.88316733584741, 14.22301141913599
