| L(s) = 1 | − 2.78·2-s − 2.83·3-s + 5.72·4-s − 1.35·5-s + 7.88·6-s + 4.56·7-s − 10.3·8-s + 5.03·9-s + 3.77·10-s − 2.32·11-s − 16.2·12-s + 0.925·13-s − 12.6·14-s + 3.84·15-s + 17.3·16-s + 6.55·17-s − 14.0·18-s + 3.74·19-s − 7.78·20-s − 12.9·21-s + 6.45·22-s − 3.65·23-s + 29.3·24-s − 3.15·25-s − 2.57·26-s − 5.77·27-s + 26.1·28-s + ⋯ |
| L(s) = 1 | − 1.96·2-s − 1.63·3-s + 2.86·4-s − 0.607·5-s + 3.21·6-s + 1.72·7-s − 3.66·8-s + 1.67·9-s + 1.19·10-s − 0.699·11-s − 4.68·12-s + 0.256·13-s − 3.39·14-s + 0.994·15-s + 4.34·16-s + 1.58·17-s − 3.30·18-s + 0.858·19-s − 1.73·20-s − 2.82·21-s + 1.37·22-s − 0.761·23-s + 6.00·24-s − 0.631·25-s − 0.504·26-s − 1.11·27-s + 4.94·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8023 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8023 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 71 | \( 1 + T \) |
| 113 | \( 1 + T \) |
| good | 2 | \( 1 + 2.78T + 2T^{2} \) |
| 3 | \( 1 + 2.83T + 3T^{2} \) |
| 5 | \( 1 + 1.35T + 5T^{2} \) |
| 7 | \( 1 - 4.56T + 7T^{2} \) |
| 11 | \( 1 + 2.32T + 11T^{2} \) |
| 13 | \( 1 - 0.925T + 13T^{2} \) |
| 17 | \( 1 - 6.55T + 17T^{2} \) |
| 19 | \( 1 - 3.74T + 19T^{2} \) |
| 23 | \( 1 + 3.65T + 23T^{2} \) |
| 29 | \( 1 - 6.80T + 29T^{2} \) |
| 31 | \( 1 - 0.567T + 31T^{2} \) |
| 37 | \( 1 + 0.527T + 37T^{2} \) |
| 41 | \( 1 + 10.2T + 41T^{2} \) |
| 43 | \( 1 - 2.28T + 43T^{2} \) |
| 47 | \( 1 - 6.93T + 47T^{2} \) |
| 53 | \( 1 + 6.32T + 53T^{2} \) |
| 59 | \( 1 - 4.25T + 59T^{2} \) |
| 61 | \( 1 - 13.6T + 61T^{2} \) |
| 67 | \( 1 - 0.248T + 67T^{2} \) |
| 73 | \( 1 + 4.69T + 73T^{2} \) |
| 79 | \( 1 + 10.4T + 79T^{2} \) |
| 83 | \( 1 + 12.1T + 83T^{2} \) |
| 89 | \( 1 + 11.0T + 89T^{2} \) |
| 97 | \( 1 + 11.9T + 97T^{2} \) |
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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
−7.48579447026840996690932145341, −7.23546638291304038270083856167, −6.21917098225623820859591933773, −5.52153372464739569888394698039, −5.11646622786527353992210942892, −3.90972823547577038829686158504, −2.67099975981658973795231291266, −1.47517326782295072814045554824, −1.06368369263331774634342495413, 0,
1.06368369263331774634342495413, 1.47517326782295072814045554824, 2.67099975981658973795231291266, 3.90972823547577038829686158504, 5.11646622786527353992210942892, 5.52153372464739569888394698039, 6.21917098225623820859591933773, 7.23546638291304038270083856167, 7.48579447026840996690932145341
