| L(s) = 1 | − 1.60·3-s − 5-s + 7-s − 0.426·9-s − 3.70·11-s − 4.52·13-s + 1.60·15-s + 3.84·17-s + 1.03·19-s − 1.60·21-s − 0.0720·23-s + 25-s + 5.49·27-s + 5.78·29-s + 1.81·31-s + 5.94·33-s − 35-s + 1.11·37-s + 7.26·39-s + 9.44·41-s − 43-s + 0.426·45-s + 6.74·47-s + 49-s − 6.16·51-s − 8.80·53-s + 3.70·55-s + ⋯ |
| L(s) = 1 | − 0.926·3-s − 0.447·5-s + 0.377·7-s − 0.142·9-s − 1.11·11-s − 1.25·13-s + 0.414·15-s + 0.931·17-s + 0.236·19-s − 0.350·21-s − 0.0150·23-s + 0.200·25-s + 1.05·27-s + 1.07·29-s + 0.325·31-s + 1.03·33-s − 0.169·35-s + 0.182·37-s + 1.16·39-s + 1.47·41-s − 0.152·43-s + 0.0635·45-s + 0.984·47-s + 0.142·49-s − 0.862·51-s − 1.20·53-s + 0.499·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 43 | \( 1 + T \) |
| good | 3 | \( 1 + 1.60T + 3T^{2} \) |
| 11 | \( 1 + 3.70T + 11T^{2} \) |
| 13 | \( 1 + 4.52T + 13T^{2} \) |
| 17 | \( 1 - 3.84T + 17T^{2} \) |
| 19 | \( 1 - 1.03T + 19T^{2} \) |
| 23 | \( 1 + 0.0720T + 23T^{2} \) |
| 29 | \( 1 - 5.78T + 29T^{2} \) |
| 31 | \( 1 - 1.81T + 31T^{2} \) |
| 37 | \( 1 - 1.11T + 37T^{2} \) |
| 41 | \( 1 - 9.44T + 41T^{2} \) |
| 47 | \( 1 - 6.74T + 47T^{2} \) |
| 53 | \( 1 + 8.80T + 53T^{2} \) |
| 59 | \( 1 + 4.68T + 59T^{2} \) |
| 61 | \( 1 + 6.50T + 61T^{2} \) |
| 67 | \( 1 + 0.238T + 67T^{2} \) |
| 71 | \( 1 + 0.589T + 71T^{2} \) |
| 73 | \( 1 - 2.62T + 73T^{2} \) |
| 79 | \( 1 - 15.0T + 79T^{2} \) |
| 83 | \( 1 - 5.63T + 83T^{2} \) |
| 89 | \( 1 + 17.3T + 89T^{2} \) |
| 97 | \( 1 - 5.76T + 97T^{2} \) |
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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
−7.84311366586142657082776208841, −7.05900916069995005683531988523, −6.19957269320190326108454408926, −5.45084878028898198201291725479, −4.97048873865086666426289479412, −4.33153524850393718862307158210, −3.06697614649736314138713783372, −2.47724057821813098802795455920, −1.01618230572415559550786860081, 0,
1.01618230572415559550786860081, 2.47724057821813098802795455920, 3.06697614649736314138713783372, 4.33153524850393718862307158210, 4.97048873865086666426289479412, 5.45084878028898198201291725479, 6.19957269320190326108454408926, 7.05900916069995005683531988523, 7.84311366586142657082776208841
