| L(s) = 1 | − 2-s + 0.195·3-s + 4-s − 1.22·5-s − 0.195·6-s + 4.43·7-s − 8-s − 2.96·9-s + 1.22·10-s + 2.56·11-s + 0.195·12-s − 6.10·13-s − 4.43·14-s − 0.240·15-s + 16-s + 2.44·17-s + 2.96·18-s − 7.58·19-s − 1.22·20-s + 0.868·21-s − 2.56·22-s − 5.74·23-s − 0.195·24-s − 3.49·25-s + 6.10·26-s − 1.16·27-s + 4.43·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.113·3-s + 0.5·4-s − 0.548·5-s − 0.0799·6-s + 1.67·7-s − 0.353·8-s − 0.987·9-s + 0.387·10-s + 0.774·11-s + 0.0565·12-s − 1.69·13-s − 1.18·14-s − 0.0620·15-s + 0.250·16-s + 0.593·17-s + 0.698·18-s − 1.73·19-s − 0.274·20-s + 0.189·21-s − 0.547·22-s − 1.19·23-s − 0.0399·24-s − 0.698·25-s + 1.19·26-s − 0.224·27-s + 0.838·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1502 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1502 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 751 | \( 1 + T \) |
| good | 3 | \( 1 - 0.195T + 3T^{2} \) |
| 5 | \( 1 + 1.22T + 5T^{2} \) |
| 7 | \( 1 - 4.43T + 7T^{2} \) |
| 11 | \( 1 - 2.56T + 11T^{2} \) |
| 13 | \( 1 + 6.10T + 13T^{2} \) |
| 17 | \( 1 - 2.44T + 17T^{2} \) |
| 19 | \( 1 + 7.58T + 19T^{2} \) |
| 23 | \( 1 + 5.74T + 23T^{2} \) |
| 29 | \( 1 - 8.59T + 29T^{2} \) |
| 31 | \( 1 + 3.25T + 31T^{2} \) |
| 37 | \( 1 + 1.07T + 37T^{2} \) |
| 41 | \( 1 + 6.77T + 41T^{2} \) |
| 43 | \( 1 - 5.01T + 43T^{2} \) |
| 47 | \( 1 + 7.81T + 47T^{2} \) |
| 53 | \( 1 - 8.11T + 53T^{2} \) |
| 59 | \( 1 - 2.41T + 59T^{2} \) |
| 61 | \( 1 + 12.4T + 61T^{2} \) |
| 67 | \( 1 - 0.828T + 67T^{2} \) |
| 71 | \( 1 + 5.18T + 71T^{2} \) |
| 73 | \( 1 - 1.45T + 73T^{2} \) |
| 79 | \( 1 - 6.32T + 79T^{2} \) |
| 83 | \( 1 + 13.6T + 83T^{2} \) |
| 89 | \( 1 + 8.91T + 89T^{2} \) |
| 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
−8.835633630046762043090198965336, −8.228221037061071091308696369547, −7.82365277527374865018843588490, −6.88564626988036370391294371091, −5.83180263064878226273986985184, −4.83723485284413282102952648148, −4.03239098445538814732889255493, −2.58084737601136772017862124147, −1.71219486929232588262523554336, 0,
1.71219486929232588262523554336, 2.58084737601136772017862124147, 4.03239098445538814732889255493, 4.83723485284413282102952648148, 5.83180263064878226273986985184, 6.88564626988036370391294371091, 7.82365277527374865018843588490, 8.228221037061071091308696369547, 8.835633630046762043090198965336
