| L(s) = 1 | + 0.892·2-s + 0.942·3-s − 1.20·4-s + 0.841·6-s + 7-s − 2.85·8-s − 2.11·9-s − 6.14·11-s − 1.13·12-s − 5.43·13-s + 0.892·14-s − 0.146·16-s + 5.42·17-s − 1.88·18-s + 0.0416·19-s + 0.942·21-s − 5.48·22-s − 3.12·23-s − 2.69·24-s − 4.85·26-s − 4.81·27-s − 1.20·28-s − 5.24·29-s + 1.05·31-s + 5.58·32-s − 5.79·33-s + 4.84·34-s + ⋯ |
| L(s) = 1 | + 0.631·2-s + 0.544·3-s − 0.601·4-s + 0.343·6-s + 0.377·7-s − 1.01·8-s − 0.704·9-s − 1.85·11-s − 0.327·12-s − 1.50·13-s + 0.238·14-s − 0.0366·16-s + 1.31·17-s − 0.444·18-s + 0.00954·19-s + 0.205·21-s − 1.16·22-s − 0.650·23-s − 0.550·24-s − 0.951·26-s − 0.927·27-s − 0.227·28-s − 0.974·29-s + 0.189·31-s + 0.987·32-s − 1.00·33-s + 0.830·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 875 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 875 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 2 | \( 1 - 0.892T + 2T^{2} \) |
| 3 | \( 1 - 0.942T + 3T^{2} \) |
| 11 | \( 1 + 6.14T + 11T^{2} \) |
| 13 | \( 1 + 5.43T + 13T^{2} \) |
| 17 | \( 1 - 5.42T + 17T^{2} \) |
| 19 | \( 1 - 0.0416T + 19T^{2} \) |
| 23 | \( 1 + 3.12T + 23T^{2} \) |
| 29 | \( 1 + 5.24T + 29T^{2} \) |
| 31 | \( 1 - 1.05T + 31T^{2} \) |
| 37 | \( 1 + 6.65T + 37T^{2} \) |
| 41 | \( 1 - 10.0T + 41T^{2} \) |
| 43 | \( 1 - 9.39T + 43T^{2} \) |
| 47 | \( 1 - 0.370T + 47T^{2} \) |
| 53 | \( 1 + 9.12T + 53T^{2} \) |
| 59 | \( 1 - 8.09T + 59T^{2} \) |
| 61 | \( 1 - 1.20T + 61T^{2} \) |
| 67 | \( 1 + 13.1T + 67T^{2} \) |
| 71 | \( 1 - 6.42T + 71T^{2} \) |
| 73 | \( 1 + 13.3T + 73T^{2} \) |
| 79 | \( 1 + 9.76T + 79T^{2} \) |
| 83 | \( 1 + 4.44T + 83T^{2} \) |
| 89 | \( 1 - 5.41T + 89T^{2} \) |
| 97 | \( 1 + 6.12T + 97T^{2} \) |
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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
−9.724916790186189025748227413661, −8.867301312985004337055289317160, −7.86203320251066036387955552952, −7.56845254891064057354605268408, −5.69335817077744701462123410571, −5.38554690890054558808365444124, −4.37671692770957831747655032413, −3.14088486473103324013327877837, −2.42877055864525188948343749559, 0,
2.42877055864525188948343749559, 3.14088486473103324013327877837, 4.37671692770957831747655032413, 5.38554690890054558808365444124, 5.69335817077744701462123410571, 7.56845254891064057354605268408, 7.86203320251066036387955552952, 8.867301312985004337055289317160, 9.724916790186189025748227413661
