| L(s) = 1 | + 2-s − 3.30·3-s + 4-s + 5-s − 3.30·6-s + 8-s + 7.90·9-s + 10-s + 11-s − 3.30·12-s + 5.30·13-s − 3.30·15-s + 16-s + 0.605·17-s + 7.90·18-s − 1.69·19-s + 20-s + 22-s − 4.60·23-s − 3.30·24-s + 25-s + 5.30·26-s − 16.2·27-s − 2.60·29-s − 3.30·30-s − 6·31-s + 32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.90·3-s + 0.5·4-s + 0.447·5-s − 1.34·6-s + 0.353·8-s + 2.63·9-s + 0.316·10-s + 0.301·11-s − 0.953·12-s + 1.47·13-s − 0.852·15-s + 0.250·16-s + 0.146·17-s + 1.86·18-s − 0.389·19-s + 0.223·20-s + 0.213·22-s − 0.960·23-s − 0.674·24-s + 0.200·25-s + 1.03·26-s − 3.11·27-s − 0.483·29-s − 0.603·30-s − 1.07·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 73 | \( 1 - T \) |
| good | 3 | \( 1 + 3.30T + 3T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 13 | \( 1 - 5.30T + 13T^{2} \) |
| 17 | \( 1 - 0.605T + 17T^{2} \) |
| 19 | \( 1 + 1.69T + 19T^{2} \) |
| 23 | \( 1 + 4.60T + 23T^{2} \) |
| 29 | \( 1 + 2.60T + 29T^{2} \) |
| 31 | \( 1 + 6T + 31T^{2} \) |
| 37 | \( 1 + 6T + 37T^{2} \) |
| 41 | \( 1 + 8.60T + 41T^{2} \) |
| 43 | \( 1 + 4.30T + 43T^{2} \) |
| 47 | \( 1 + 8.90T + 47T^{2} \) |
| 53 | \( 1 - 12.6T + 53T^{2} \) |
| 59 | \( 1 - 1.30T + 59T^{2} \) |
| 61 | \( 1 - 1.90T + 61T^{2} \) |
| 67 | \( 1 + 1.90T + 67T^{2} \) |
| 71 | \( 1 - 3.51T + 71T^{2} \) |
| 79 | \( 1 + 2T + 79T^{2} \) |
| 83 | \( 1 + 5.30T + 83T^{2} \) |
| 89 | \( 1 + 2.09T + 89T^{2} \) |
| 97 | \( 1 + 7.39T + 97T^{2} \) |
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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
−6.93616776631957037828854511239, −6.62431838620476628122699339458, −5.95671400003018670379256591750, −5.49456113919515675514798592581, −4.91731215793884481566086808525, −4.00162469362272215774231811981, −3.52700469345695590795194804216, −1.88305386638260989342689475200, −1.34522589133823824732131733330, 0,
1.34522589133823824732131733330, 1.88305386638260989342689475200, 3.52700469345695590795194804216, 4.00162469362272215774231811981, 4.91731215793884481566086808525, 5.49456113919515675514798592581, 5.95671400003018670379256591750, 6.62431838620476628122699339458, 6.93616776631957037828854511239
