| L(s) = 1 | + 3-s − 5-s + 3.15·7-s + 9-s + 2.98·11-s + 0.285·13-s − 15-s + 3.60·17-s − 0.827·19-s + 3.15·21-s + 8.54·23-s + 25-s + 27-s − 7.01·29-s − 5.11·31-s + 2.98·33-s − 3.15·35-s + 7.86·37-s + 0.285·39-s − 0.299·41-s − 0.325·43-s − 45-s + 6.94·47-s + 2.95·49-s + 3.60·51-s − 8.00·53-s − 2.98·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s + 1.19·7-s + 0.333·9-s + 0.900·11-s + 0.0792·13-s − 0.258·15-s + 0.875·17-s − 0.189·19-s + 0.688·21-s + 1.78·23-s + 0.200·25-s + 0.192·27-s − 1.30·29-s − 0.919·31-s + 0.520·33-s − 0.533·35-s + 1.29·37-s + 0.0457·39-s − 0.0467·41-s − 0.0495·43-s − 0.149·45-s + 1.01·47-s + 0.421·49-s + 0.505·51-s − 1.09·53-s − 0.402·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.930679113\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.930679113\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 67 | \( 1 - T \) |
| good | 7 | \( 1 - 3.15T + 7T^{2} \) |
| 11 | \( 1 - 2.98T + 11T^{2} \) |
| 13 | \( 1 - 0.285T + 13T^{2} \) |
| 17 | \( 1 - 3.60T + 17T^{2} \) |
| 19 | \( 1 + 0.827T + 19T^{2} \) |
| 23 | \( 1 - 8.54T + 23T^{2} \) |
| 29 | \( 1 + 7.01T + 29T^{2} \) |
| 31 | \( 1 + 5.11T + 31T^{2} \) |
| 37 | \( 1 - 7.86T + 37T^{2} \) |
| 41 | \( 1 + 0.299T + 41T^{2} \) |
| 43 | \( 1 + 0.325T + 43T^{2} \) |
| 47 | \( 1 - 6.94T + 47T^{2} \) |
| 53 | \( 1 + 8.00T + 53T^{2} \) |
| 59 | \( 1 + 7.59T + 59T^{2} \) |
| 61 | \( 1 + 7.05T + 61T^{2} \) |
| 71 | \( 1 - 14.9T + 71T^{2} \) |
| 73 | \( 1 + 5.66T + 73T^{2} \) |
| 79 | \( 1 + 1.40T + 79T^{2} \) |
| 83 | \( 1 - 4.16T + 83T^{2} \) |
| 89 | \( 1 - 4.37T + 89T^{2} \) |
| 97 | \( 1 - 3.52T + 97T^{2} \) |
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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
−8.417941048303536983174951407218, −7.65323536006829534968439276180, −7.31525433803497543963843159536, −6.29226617446277251324228236913, −5.32356103120147850382065552113, −4.60339636847024588774997001690, −3.82187588336688926904958804231, −3.05910331330439458177127636169, −1.86769401053889291804268499106, −1.04609504354782529066400957515,
1.04609504354782529066400957515, 1.86769401053889291804268499106, 3.05910331330439458177127636169, 3.82187588336688926904958804231, 4.60339636847024588774997001690, 5.32356103120147850382065552113, 6.29226617446277251324228236913, 7.31525433803497543963843159536, 7.65323536006829534968439276180, 8.417941048303536983174951407218
