| L(s) = 1 | − 2.59·2-s + 4.71·4-s − 3.45·5-s − 7-s − 7.04·8-s + 8.95·10-s + 0.590·11-s − 0.545·13-s + 2.59·14-s + 8.81·16-s + 5.73·17-s + 0.427·19-s − 16.2·20-s − 1.52·22-s + 2.92·23-s + 6.93·25-s + 1.41·26-s − 4.71·28-s − 1.28·29-s + 1.26·31-s − 8.76·32-s − 14.8·34-s + 3.45·35-s − 9.04·37-s − 1.10·38-s + 24.3·40-s − 2.37·41-s + ⋯ |
| L(s) = 1 | − 1.83·2-s + 2.35·4-s − 1.54·5-s − 0.377·7-s − 2.48·8-s + 2.83·10-s + 0.177·11-s − 0.151·13-s + 0.692·14-s + 2.20·16-s + 1.39·17-s + 0.0980·19-s − 3.64·20-s − 0.326·22-s + 0.609·23-s + 1.38·25-s + 0.277·26-s − 0.891·28-s − 0.237·29-s + 0.227·31-s − 1.54·32-s − 2.54·34-s + 0.583·35-s − 1.48·37-s − 0.179·38-s + 3.84·40-s − 0.371·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 127 | \( 1 - T \) |
| good | 2 | \( 1 + 2.59T + 2T^{2} \) |
| 5 | \( 1 + 3.45T + 5T^{2} \) |
| 11 | \( 1 - 0.590T + 11T^{2} \) |
| 13 | \( 1 + 0.545T + 13T^{2} \) |
| 17 | \( 1 - 5.73T + 17T^{2} \) |
| 19 | \( 1 - 0.427T + 19T^{2} \) |
| 23 | \( 1 - 2.92T + 23T^{2} \) |
| 29 | \( 1 + 1.28T + 29T^{2} \) |
| 31 | \( 1 - 1.26T + 31T^{2} \) |
| 37 | \( 1 + 9.04T + 37T^{2} \) |
| 41 | \( 1 + 2.37T + 41T^{2} \) |
| 43 | \( 1 + 5.15T + 43T^{2} \) |
| 47 | \( 1 + 12.5T + 47T^{2} \) |
| 53 | \( 1 - 5.31T + 53T^{2} \) |
| 59 | \( 1 + 9.87T + 59T^{2} \) |
| 61 | \( 1 - 5.02T + 61T^{2} \) |
| 67 | \( 1 - 2.64T + 67T^{2} \) |
| 71 | \( 1 + 13.1T + 71T^{2} \) |
| 73 | \( 1 - 15.5T + 73T^{2} \) |
| 79 | \( 1 + 7.77T + 79T^{2} \) |
| 83 | \( 1 - 3.24T + 83T^{2} \) |
| 89 | \( 1 - 6.62T + 89T^{2} \) |
| 97 | \( 1 + 3.67T + 97T^{2} \) |
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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
−7.68436565630603297999342587846, −7.08188953456979135391085136155, −6.63991031402456564134006006401, −5.60414192312228756206861643847, −4.63440233776314494377486146237, −3.37401857754731667227488296001, −3.21997259735021446531074192601, −1.84881910490613130034532373051, −0.877335465664038184430943886279, 0,
0.877335465664038184430943886279, 1.84881910490613130034532373051, 3.21997259735021446531074192601, 3.37401857754731667227488296001, 4.63440233776314494377486146237, 5.60414192312228756206861643847, 6.63991031402456564134006006401, 7.08188953456979135391085136155, 7.68436565630603297999342587846
