| L(s) = 1 | + 0.343·2-s − 1.88·4-s + 4.21·5-s + 7-s − 1.33·8-s + 1.44·10-s + 0.252·11-s − 6.96·13-s + 0.343·14-s + 3.30·16-s − 2.67·17-s − 7.42·19-s − 7.93·20-s + 0.0866·22-s − 0.137·23-s + 12.7·25-s − 2.39·26-s − 1.88·28-s + 2.53·29-s + 2.20·31-s + 3.80·32-s − 0.918·34-s + 4.21·35-s + 7.40·37-s − 2.55·38-s − 5.62·40-s + 4.80·41-s + ⋯ |
| L(s) = 1 | + 0.242·2-s − 0.940·4-s + 1.88·5-s + 0.377·7-s − 0.471·8-s + 0.457·10-s + 0.0760·11-s − 1.93·13-s + 0.0918·14-s + 0.826·16-s − 0.648·17-s − 1.70·19-s − 1.77·20-s + 0.0184·22-s − 0.0287·23-s + 2.55·25-s − 0.468·26-s − 0.355·28-s + 0.470·29-s + 0.395·31-s + 0.672·32-s − 0.157·34-s + 0.712·35-s + 1.21·37-s − 0.414·38-s − 0.888·40-s + 0.750·41-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 | \( 1 - 0.343T + 2T^{2} \) |
| 5 | \( 1 - 4.21T + 5T^{2} \) |
| 11 | \( 1 - 0.252T + 11T^{2} \) |
| 13 | \( 1 + 6.96T + 13T^{2} \) |
| 17 | \( 1 + 2.67T + 17T^{2} \) |
| 19 | \( 1 + 7.42T + 19T^{2} \) |
| 23 | \( 1 + 0.137T + 23T^{2} \) |
| 29 | \( 1 - 2.53T + 29T^{2} \) |
| 31 | \( 1 - 2.20T + 31T^{2} \) |
| 37 | \( 1 - 7.40T + 37T^{2} \) |
| 41 | \( 1 - 4.80T + 41T^{2} \) |
| 43 | \( 1 - 2.71T + 43T^{2} \) |
| 47 | \( 1 + 5.23T + 47T^{2} \) |
| 53 | \( 1 + 7.52T + 53T^{2} \) |
| 59 | \( 1 + 5.54T + 59T^{2} \) |
| 61 | \( 1 - 0.883T + 61T^{2} \) |
| 67 | \( 1 + 15.1T + 67T^{2} \) |
| 71 | \( 1 + 6.41T + 71T^{2} \) |
| 73 | \( 1 - 5.39T + 73T^{2} \) |
| 79 | \( 1 + 9.73T + 79T^{2} \) |
| 83 | \( 1 - 4.76T + 83T^{2} \) |
| 89 | \( 1 - 7.64T + 89T^{2} \) |
| 97 | \( 1 + 0.551T + 97T^{2} \) |
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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
−7.50619712668646391463747271918, −6.46359566996438751501475711603, −6.14094044501542440328884230168, −5.27167321638358615668496109736, −4.70547908601295875465058547706, −4.30081944431560085661743566014, −2.77519299206508591108161449012, −2.37040464177359392379170880751, −1.40205457009655107181472856018, 0,
1.40205457009655107181472856018, 2.37040464177359392379170880751, 2.77519299206508591108161449012, 4.30081944431560085661743566014, 4.70547908601295875465058547706, 5.27167321638358615668496109736, 6.14094044501542440328884230168, 6.46359566996438751501475711603, 7.50619712668646391463747271918
