| L(s) = 1 | + 0.618·3-s − 4.85·7-s − 2.61·9-s + 3.38·11-s + 0.381·13-s − 5.85·17-s + 6.85·19-s − 3.00·21-s − 23-s − 3.47·27-s + 3.70·29-s + 8.85·31-s + 2.09·33-s + 3.70·37-s + 0.236·39-s − 3.38·41-s − 6.76·43-s + 11.7·47-s + 16.5·49-s − 3.61·51-s − 2·53-s + 4.23·57-s + 6·59-s − 3.85·61-s + 12.7·63-s − 0.763·67-s − 0.618·69-s + ⋯ |
| L(s) = 1 | + 0.356·3-s − 1.83·7-s − 0.872·9-s + 1.01·11-s + 0.105·13-s − 1.41·17-s + 1.57·19-s − 0.654·21-s − 0.208·23-s − 0.668·27-s + 0.688·29-s + 1.59·31-s + 0.363·33-s + 0.609·37-s + 0.0378·39-s − 0.528·41-s − 1.03·43-s + 1.70·47-s + 2.36·49-s − 0.506·51-s − 0.274·53-s + 0.561·57-s + 0.781·59-s − 0.493·61-s + 1.60·63-s − 0.0933·67-s − 0.0744·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 3 | \( 1 - 0.618T + 3T^{2} \) |
| 7 | \( 1 + 4.85T + 7T^{2} \) |
| 11 | \( 1 - 3.38T + 11T^{2} \) |
| 13 | \( 1 - 0.381T + 13T^{2} \) |
| 17 | \( 1 + 5.85T + 17T^{2} \) |
| 19 | \( 1 - 6.85T + 19T^{2} \) |
| 29 | \( 1 - 3.70T + 29T^{2} \) |
| 31 | \( 1 - 8.85T + 31T^{2} \) |
| 37 | \( 1 - 3.70T + 37T^{2} \) |
| 41 | \( 1 + 3.38T + 41T^{2} \) |
| 43 | \( 1 + 6.76T + 43T^{2} \) |
| 47 | \( 1 - 11.7T + 47T^{2} \) |
| 53 | \( 1 + 2T + 53T^{2} \) |
| 59 | \( 1 - 6T + 59T^{2} \) |
| 61 | \( 1 + 3.85T + 61T^{2} \) |
| 67 | \( 1 + 0.763T + 67T^{2} \) |
| 71 | \( 1 + 2.61T + 71T^{2} \) |
| 73 | \( 1 + 7.52T + 73T^{2} \) |
| 79 | \( 1 + 5.70T + 79T^{2} \) |
| 83 | \( 1 - 5.70T + 83T^{2} \) |
| 89 | \( 1 + 9.70T + 89T^{2} \) |
| 97 | \( 1 + 16.0T + 97T^{2} \) |
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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
−7.18813099154610593373264390520, −6.64012365575618270497477040768, −6.18037653717170611055521593103, −5.48671076403319325537337688192, −4.40207909649508648430385415823, −3.71419787142651379007852250158, −2.96689212578391066925077618404, −2.54466105114716334247328434436, −1.11342017386227812553296594761, 0,
1.11342017386227812553296594761, 2.54466105114716334247328434436, 2.96689212578391066925077618404, 3.71419787142651379007852250158, 4.40207909649508648430385415823, 5.48671076403319325537337688192, 6.18037653717170611055521593103, 6.64012365575618270497477040768, 7.18813099154610593373264390520
