| L(s) = 1 | + 1.62·3-s − 0.534·5-s − 2.49·7-s − 0.356·9-s + 4.48·11-s − 3.62·13-s − 0.868·15-s + 1.13·17-s − 8.04·19-s − 4.05·21-s − 0.329·23-s − 4.71·25-s − 5.45·27-s + 0.0975·29-s − 31-s + 7.28·33-s + 1.33·35-s − 9.95·37-s − 5.89·39-s + 11.0·41-s + 0.899·43-s + 0.190·45-s + 0.827·47-s − 0.785·49-s + 1.83·51-s + 11.1·53-s − 2.39·55-s + ⋯ |
| L(s) = 1 | + 0.938·3-s − 0.238·5-s − 0.942·7-s − 0.118·9-s + 1.35·11-s − 1.00·13-s − 0.224·15-s + 0.274·17-s − 1.84·19-s − 0.884·21-s − 0.0687·23-s − 0.942·25-s − 1.05·27-s + 0.0181·29-s − 0.179·31-s + 1.26·33-s + 0.225·35-s − 1.63·37-s − 0.943·39-s + 1.72·41-s + 0.137·43-s + 0.0283·45-s + 0.120·47-s − 0.112·49-s + 0.257·51-s + 1.53·53-s − 0.322·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1984 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1984 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 31 | \( 1 + T \) |
| good | 3 | \( 1 - 1.62T + 3T^{2} \) |
| 5 | \( 1 + 0.534T + 5T^{2} \) |
| 7 | \( 1 + 2.49T + 7T^{2} \) |
| 11 | \( 1 - 4.48T + 11T^{2} \) |
| 13 | \( 1 + 3.62T + 13T^{2} \) |
| 17 | \( 1 - 1.13T + 17T^{2} \) |
| 19 | \( 1 + 8.04T + 19T^{2} \) |
| 23 | \( 1 + 0.329T + 23T^{2} \) |
| 29 | \( 1 - 0.0975T + 29T^{2} \) |
| 37 | \( 1 + 9.95T + 37T^{2} \) |
| 41 | \( 1 - 11.0T + 41T^{2} \) |
| 43 | \( 1 - 0.899T + 43T^{2} \) |
| 47 | \( 1 - 0.827T + 47T^{2} \) |
| 53 | \( 1 - 11.1T + 53T^{2} \) |
| 59 | \( 1 + 8.58T + 59T^{2} \) |
| 61 | \( 1 + 11.7T + 61T^{2} \) |
| 67 | \( 1 + 3.78T + 67T^{2} \) |
| 71 | \( 1 + 5.66T + 71T^{2} \) |
| 73 | \( 1 - 12.1T + 73T^{2} \) |
| 79 | \( 1 - 0.649T + 79T^{2} \) |
| 83 | \( 1 + 14.4T + 83T^{2} \) |
| 89 | \( 1 - 1.45T + 89T^{2} \) |
| 97 | \( 1 + 4.79T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.959412778707080744600733524779, −8.077792602955466144408712500325, −7.26624309517400907518164747744, −6.48553310558116664735385305313, −5.74400170371314486819499925181, −4.33853407210144816713096520414, −3.74541101489899217903625445607, −2.82067921508041123239430536491, −1.89026299791377698862754546946, 0,
1.89026299791377698862754546946, 2.82067921508041123239430536491, 3.74541101489899217903625445607, 4.33853407210144816713096520414, 5.74400170371314486819499925181, 6.48553310558116664735385305313, 7.26624309517400907518164747744, 8.077792602955466144408712500325, 8.959412778707080744600733524779
