| L(s) = 1 | − 2.65·2-s − 3-s + 5.04·4-s + 0.115·5-s + 2.65·6-s + 1.37·7-s − 8.07·8-s + 9-s − 0.305·10-s − 1.70·11-s − 5.04·12-s − 3.57·13-s − 3.64·14-s − 0.115·15-s + 11.3·16-s − 3.93·17-s − 2.65·18-s + 3.95·19-s + 0.581·20-s − 1.37·21-s + 4.51·22-s + 2.56·23-s + 8.07·24-s − 4.98·25-s + 9.49·26-s − 27-s + 6.91·28-s + ⋯ |
| L(s) = 1 | − 1.87·2-s − 0.577·3-s + 2.52·4-s + 0.0515·5-s + 1.08·6-s + 0.518·7-s − 2.85·8-s + 0.333·9-s − 0.0967·10-s − 0.513·11-s − 1.45·12-s − 0.992·13-s − 0.973·14-s − 0.0297·15-s + 2.83·16-s − 0.954·17-s − 0.625·18-s + 0.907·19-s + 0.130·20-s − 0.299·21-s + 0.962·22-s + 0.533·23-s + 1.64·24-s − 0.997·25-s + 1.86·26-s − 0.192·27-s + 1.30·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 633 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 633 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 211 | \( 1 + T \) |
| good | 2 | \( 1 + 2.65T + 2T^{2} \) |
| 5 | \( 1 - 0.115T + 5T^{2} \) |
| 7 | \( 1 - 1.37T + 7T^{2} \) |
| 11 | \( 1 + 1.70T + 11T^{2} \) |
| 13 | \( 1 + 3.57T + 13T^{2} \) |
| 17 | \( 1 + 3.93T + 17T^{2} \) |
| 19 | \( 1 - 3.95T + 19T^{2} \) |
| 23 | \( 1 - 2.56T + 23T^{2} \) |
| 29 | \( 1 - 2.38T + 29T^{2} \) |
| 31 | \( 1 + 0.603T + 31T^{2} \) |
| 37 | \( 1 - 1.70T + 37T^{2} \) |
| 41 | \( 1 - 2.35T + 41T^{2} \) |
| 43 | \( 1 + 1.12T + 43T^{2} \) |
| 47 | \( 1 + 8.96T + 47T^{2} \) |
| 53 | \( 1 + 0.866T + 53T^{2} \) |
| 59 | \( 1 + 2.20T + 59T^{2} \) |
| 61 | \( 1 + 2.85T + 61T^{2} \) |
| 67 | \( 1 + 13.3T + 67T^{2} \) |
| 71 | \( 1 + 9.66T + 71T^{2} \) |
| 73 | \( 1 - 13.2T + 73T^{2} \) |
| 79 | \( 1 + 14.2T + 79T^{2} \) |
| 83 | \( 1 + 13.8T + 83T^{2} \) |
| 89 | \( 1 - 0.748T + 89T^{2} \) |
| 97 | \( 1 + 18.5T + 97T^{2} \) |
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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
−9.990882943622309512605804457137, −9.448643062344913114094839757890, −8.424627540756676612343479165591, −7.62317699394459736461063224608, −6.98750022935890161399546952855, −5.93273922036629033462161925268, −4.77592312393734414998275556219, −2.77999891390825398728535280572, −1.58483181713402509564598743711, 0,
1.58483181713402509564598743711, 2.77999891390825398728535280572, 4.77592312393734414998275556219, 5.93273922036629033462161925268, 6.98750022935890161399546952855, 7.62317699394459736461063224608, 8.424627540756676612343479165591, 9.448643062344913114094839757890, 9.990882943622309512605804457137
