| L(s) = 1 | − 7-s + 11-s − 4·13-s − 2·17-s − 4·19-s + 2·23-s + 2·31-s + 8·37-s + 6·41-s + 4·43-s + 49-s − 6·53-s + 4·59-s + 6·61-s + 4·67-s + 4·73-s − 77-s + 14·79-s + 6·83-s + 4·91-s + 10·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + 2·119-s + ⋯ |
| L(s) = 1 | − 0.377·7-s + 0.301·11-s − 1.10·13-s − 0.485·17-s − 0.917·19-s + 0.417·23-s + 0.359·31-s + 1.31·37-s + 0.937·41-s + 0.609·43-s + 1/7·49-s − 0.824·53-s + 0.520·59-s + 0.768·61-s + 0.488·67-s + 0.468·73-s − 0.113·77-s + 1.57·79-s + 0.658·83-s + 0.419·91-s + 1.01·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + 0.183·119-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 277200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 277200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.178273235\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.178273235\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| good | 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 14 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−12.75227965848139, −12.40015042133777, −11.88581616231735, −11.27816727631987, −11.03270183880614, −10.44586643650281, −9.919306092730823, −9.551324882913511, −9.124050286140111, −8.633090479090326, −8.096701685196378, −7.532012806810015, −7.209824334063312, −6.510025396047062, −6.263608666882597, −5.732272396040559, −4.855646195615932, −4.777351724257280, −4.038219118020308, −3.605733651189557, −2.834514424975515, −2.365780051081284, −1.974313946186011, −0.9492748956348768, −0.4641289036987106,
0.4641289036987106, 0.9492748956348768, 1.974313946186011, 2.365780051081284, 2.834514424975515, 3.605733651189557, 4.038219118020308, 4.777351724257280, 4.855646195615932, 5.732272396040559, 6.263608666882597, 6.510025396047062, 7.209824334063312, 7.532012806810015, 8.096701685196378, 8.633090479090326, 9.124050286140111, 9.551324882913511, 9.919306092730823, 10.44586643650281, 11.03270183880614, 11.27816727631987, 11.88581616231735, 12.40015042133777, 12.75227965848139
