| L(s) = 1 | + 2-s − 1.80·3-s + 4-s − 3.10·5-s − 1.80·6-s − 2.67·7-s + 8-s + 0.270·9-s − 3.10·10-s + 3.86·11-s − 1.80·12-s + 1.67·13-s − 2.67·14-s + 5.60·15-s + 16-s − 6.69·17-s + 0.270·18-s + 4.35·19-s − 3.10·20-s + 4.84·21-s + 3.86·22-s + 3.62·23-s − 1.80·24-s + 4.62·25-s + 1.67·26-s + 4.93·27-s − 2.67·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.04·3-s + 0.5·4-s − 1.38·5-s − 0.738·6-s − 1.01·7-s + 0.353·8-s + 0.0902·9-s − 0.980·10-s + 1.16·11-s − 0.522·12-s + 0.465·13-s − 0.715·14-s + 1.44·15-s + 0.250·16-s − 1.62·17-s + 0.0638·18-s + 0.998·19-s − 0.693·20-s + 1.05·21-s + 0.823·22-s + 0.755·23-s − 0.369·24-s + 0.924·25-s + 0.329·26-s + 0.949·27-s − 0.506·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 2011 | \( 1 - T \) |
| good | 3 | \( 1 + 1.80T + 3T^{2} \) |
| 5 | \( 1 + 3.10T + 5T^{2} \) |
| 7 | \( 1 + 2.67T + 7T^{2} \) |
| 11 | \( 1 - 3.86T + 11T^{2} \) |
| 13 | \( 1 - 1.67T + 13T^{2} \) |
| 17 | \( 1 + 6.69T + 17T^{2} \) |
| 19 | \( 1 - 4.35T + 19T^{2} \) |
| 23 | \( 1 - 3.62T + 23T^{2} \) |
| 29 | \( 1 - 4.97T + 29T^{2} \) |
| 31 | \( 1 - 7.10T + 31T^{2} \) |
| 37 | \( 1 + 1.15T + 37T^{2} \) |
| 41 | \( 1 - 11.5T + 41T^{2} \) |
| 43 | \( 1 + 6.78T + 43T^{2} \) |
| 47 | \( 1 + 8.66T + 47T^{2} \) |
| 53 | \( 1 + 8.94T + 53T^{2} \) |
| 59 | \( 1 + 7.37T + 59T^{2} \) |
| 61 | \( 1 + 15.5T + 61T^{2} \) |
| 67 | \( 1 + 2.00T + 67T^{2} \) |
| 71 | \( 1 - 3.65T + 71T^{2} \) |
| 73 | \( 1 + 13.5T + 73T^{2} \) |
| 79 | \( 1 - 3.91T + 79T^{2} \) |
| 83 | \( 1 + 2.18T + 83T^{2} \) |
| 89 | \( 1 - 15.0T + 89T^{2} \) |
| 97 | \( 1 - 8.84T + 97T^{2} \) |
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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
−7.909219728992741638731770896346, −7.01093110308780257818298639745, −6.38110366215269213107158219037, −6.16076406879338169890028160875, −4.76985407965475980766698595831, −4.49617754190302228133045567846, −3.46986978813433057935157540702, −2.93150811708420138356954548931, −1.14627448418336538603797456371, 0,
1.14627448418336538603797456371, 2.93150811708420138356954548931, 3.46986978813433057935157540702, 4.49617754190302228133045567846, 4.76985407965475980766698595831, 6.16076406879338169890028160875, 6.38110366215269213107158219037, 7.01093110308780257818298639745, 7.909219728992741638731770896346
