| L(s) = 1 | + 5-s − 4·7-s − 3·9-s + 4·11-s + 2·13-s + 2·17-s + 4·23-s + 25-s + 2·29-s + 8·31-s − 4·35-s − 6·37-s + 6·41-s − 8·43-s − 3·45-s + 4·47-s + 9·49-s − 6·53-s + 4·55-s + 4·59-s − 2·61-s + 12·63-s + 2·65-s − 8·67-s − 6·73-s − 16·77-s + 9·81-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 1.51·7-s − 9-s + 1.20·11-s + 0.554·13-s + 0.485·17-s + 0.834·23-s + 1/5·25-s + 0.371·29-s + 1.43·31-s − 0.676·35-s − 0.986·37-s + 0.937·41-s − 1.21·43-s − 0.447·45-s + 0.583·47-s + 9/7·49-s − 0.824·53-s + 0.539·55-s + 0.520·59-s − 0.256·61-s + 1.51·63-s + 0.248·65-s − 0.977·67-s − 0.702·73-s − 1.82·77-s + 81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.852675041\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.852675041\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−16.01834844107203, −15.79369781207075, −14.81941834089751, −14.49818782444304, −13.70454190265216, −13.49432355955784, −12.75879992897925, −12.10764952240674, −11.74833604164018, −10.98248477769675, −10.32391708569410, −9.780607426043196, −9.176438111259189, −8.794771996546524, −8.147489395009597, −7.115740924191352, −6.589096062280285, −6.130684362849758, −5.635390959821321, −4.717897572878803, −3.827732313210896, −3.149846469422711, −2.757727443138700, −1.526376489663944, −0.6165339779936036,
0.6165339779936036, 1.526376489663944, 2.757727443138700, 3.149846469422711, 3.827732313210896, 4.717897572878803, 5.635390959821321, 6.130684362849758, 6.589096062280285, 7.115740924191352, 8.147489395009597, 8.794771996546524, 9.176438111259189, 9.780607426043196, 10.32391708569410, 10.98248477769675, 11.74833604164018, 12.10764952240674, 12.75879992897925, 13.49432355955784, 13.70454190265216, 14.49818782444304, 14.81941834089751, 15.79369781207075, 16.01834844107203
