| L(s) = 1 | − 2-s − 2.79·3-s + 4-s − 2.34·5-s + 2.79·6-s − 1.28·7-s − 8-s + 4.80·9-s + 2.34·10-s + 5.75·11-s − 2.79·12-s − 0.304·13-s + 1.28·14-s + 6.54·15-s + 16-s + 4.18·17-s − 4.80·18-s − 2.34·20-s + 3.58·21-s − 5.75·22-s − 6.47·23-s + 2.79·24-s + 0.497·25-s + 0.304·26-s − 5.04·27-s − 1.28·28-s + 3.12·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.61·3-s + 0.5·4-s − 1.04·5-s + 1.14·6-s − 0.485·7-s − 0.353·8-s + 1.60·9-s + 0.741·10-s + 1.73·11-s − 0.806·12-s − 0.0843·13-s + 0.343·14-s + 1.69·15-s + 0.250·16-s + 1.01·17-s − 1.13·18-s − 0.524·20-s + 0.782·21-s − 1.22·22-s − 1.35·23-s + 0.570·24-s + 0.0994·25-s + 0.0596·26-s − 0.970·27-s − 0.242·28-s + 0.580·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 + 2.79T + 3T^{2} \) |
| 5 | \( 1 + 2.34T + 5T^{2} \) |
| 7 | \( 1 + 1.28T + 7T^{2} \) |
| 11 | \( 1 - 5.75T + 11T^{2} \) |
| 13 | \( 1 + 0.304T + 13T^{2} \) |
| 17 | \( 1 - 4.18T + 17T^{2} \) |
| 23 | \( 1 + 6.47T + 23T^{2} \) |
| 29 | \( 1 - 3.12T + 29T^{2} \) |
| 31 | \( 1 + 6.44T + 31T^{2} \) |
| 37 | \( 1 - 3.97T + 37T^{2} \) |
| 41 | \( 1 - 5.01T + 41T^{2} \) |
| 43 | \( 1 + 0.989T + 43T^{2} \) |
| 47 | \( 1 + 4.39T + 47T^{2} \) |
| 53 | \( 1 + 3.29T + 53T^{2} \) |
| 59 | \( 1 + 3.31T + 59T^{2} \) |
| 61 | \( 1 + 10.9T + 61T^{2} \) |
| 67 | \( 1 - 4.38T + 67T^{2} \) |
| 71 | \( 1 - 4.41T + 71T^{2} \) |
| 73 | \( 1 + 2.26T + 73T^{2} \) |
| 79 | \( 1 + 8.57T + 79T^{2} \) |
| 83 | \( 1 + 9.76T + 83T^{2} \) |
| 89 | \( 1 - 15.0T + 89T^{2} \) |
| 97 | \( 1 + 17.3T + 97T^{2} \) |
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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
−10.00145599627902110158431853334, −9.370034991594865017601644426718, −8.125546927129838533570999708001, −7.26396626986954964448708158247, −6.42471364534263534342362214620, −5.81555434817150224211235426915, −4.43222437249828005509018297848, −3.55182529587266362830914078372, −1.32043612764107317324511906113, 0,
1.32043612764107317324511906113, 3.55182529587266362830914078372, 4.43222437249828005509018297848, 5.81555434817150224211235426915, 6.42471364534263534342362214620, 7.26396626986954964448708158247, 8.125546927129838533570999708001, 9.370034991594865017601644426718, 10.00145599627902110158431853334
