L(s) = 1 | + 0.167·3-s + 4.02·5-s − 2.97·9-s − 3.02·11-s − 2.78·13-s + 0.673·15-s + 5.41·17-s − 5.49·19-s − 23-s + 11.2·25-s − 0.998·27-s + 1.04·29-s − 7.77·31-s − 0.505·33-s − 1.57·37-s − 0.465·39-s + 0.822·41-s − 1.02·43-s − 11.9·45-s + 2.32·47-s + 0.905·51-s + 2.72·53-s − 12.1·55-s − 0.918·57-s − 10.2·59-s + 12.9·61-s − 11.2·65-s + ⋯ |
L(s) = 1 | + 0.0965·3-s + 1.80·5-s − 0.990·9-s − 0.912·11-s − 0.771·13-s + 0.174·15-s + 1.31·17-s − 1.26·19-s − 0.208·23-s + 2.24·25-s − 0.192·27-s + 0.194·29-s − 1.39·31-s − 0.0880·33-s − 0.259·37-s − 0.0745·39-s + 0.128·41-s − 0.156·43-s − 1.78·45-s + 0.339·47-s + 0.126·51-s + 0.374·53-s − 1.64·55-s − 0.121·57-s − 1.33·59-s + 1.65·61-s − 1.39·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| 23 | \( 1 + T \) |
good | 3 | \( 1 - 0.167T + 3T^{2} \) |
| 5 | \( 1 - 4.02T + 5T^{2} \) |
| 11 | \( 1 + 3.02T + 11T^{2} \) |
| 13 | \( 1 + 2.78T + 13T^{2} \) |
| 17 | \( 1 - 5.41T + 17T^{2} \) |
| 19 | \( 1 + 5.49T + 19T^{2} \) |
| 29 | \( 1 - 1.04T + 29T^{2} \) |
| 31 | \( 1 + 7.77T + 31T^{2} \) |
| 37 | \( 1 + 1.57T + 37T^{2} \) |
| 41 | \( 1 - 0.822T + 41T^{2} \) |
| 43 | \( 1 + 1.02T + 43T^{2} \) |
| 47 | \( 1 - 2.32T + 47T^{2} \) |
| 53 | \( 1 - 2.72T + 53T^{2} \) |
| 59 | \( 1 + 10.2T + 59T^{2} \) |
| 61 | \( 1 - 12.9T + 61T^{2} \) |
| 67 | \( 1 + 4.55T + 67T^{2} \) |
| 71 | \( 1 - 10.5T + 71T^{2} \) |
| 73 | \( 1 + 4.99T + 73T^{2} \) |
| 79 | \( 1 + 0.476T + 79T^{2} \) |
| 83 | \( 1 + 2.91T + 83T^{2} \) |
| 89 | \( 1 - 1.64T + 89T^{2} \) |
| 97 | \( 1 + 16.3T + 97T^{2} \) |
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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
−7.38955665613052005021632442339, −6.56156454649843797797276019409, −5.87634352452109302285599844722, −5.41104097667548727369383171754, −4.97133178118067695308536406140, −3.72868591182025129662642262802, −2.68257862203009348553431496746, −2.38141950949363110432640830945, −1.44315569413327156948054261118, 0,
1.44315569413327156948054261118, 2.38141950949363110432640830945, 2.68257862203009348553431496746, 3.72868591182025129662642262802, 4.97133178118067695308536406140, 5.41104097667548727369383171754, 5.87634352452109302285599844722, 6.56156454649843797797276019409, 7.38955665613052005021632442339