| L(s) = 1 | + 2.44·2-s − 3-s + 3.99·4-s − 4.01·5-s − 2.44·6-s + 4.89·8-s + 9-s − 9.84·10-s + 1.39·11-s − 3.99·12-s + 3.69·13-s + 4.01·15-s + 3.98·16-s + 0.554·17-s + 2.44·18-s − 6.06·19-s − 16.0·20-s + 3.40·22-s − 2.40·23-s − 4.89·24-s + 11.1·25-s + 9.05·26-s − 27-s + 0.267·29-s + 9.84·30-s + 1.40·31-s − 0.0224·32-s + ⋯ |
| L(s) = 1 | + 1.73·2-s − 0.577·3-s + 1.99·4-s − 1.79·5-s − 0.999·6-s + 1.72·8-s + 0.333·9-s − 3.11·10-s + 0.419·11-s − 1.15·12-s + 1.02·13-s + 1.03·15-s + 0.996·16-s + 0.134·17-s + 0.577·18-s − 1.39·19-s − 3.59·20-s + 0.726·22-s − 0.501·23-s − 0.998·24-s + 2.22·25-s + 1.77·26-s − 0.192·27-s + 0.0495·29-s + 1.79·30-s + 0.253·31-s − 0.00397·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| 41 | \( 1 - T \) |
| good | 2 | \( 1 - 2.44T + 2T^{2} \) |
| 5 | \( 1 + 4.01T + 5T^{2} \) |
| 11 | \( 1 - 1.39T + 11T^{2} \) |
| 13 | \( 1 - 3.69T + 13T^{2} \) |
| 17 | \( 1 - 0.554T + 17T^{2} \) |
| 19 | \( 1 + 6.06T + 19T^{2} \) |
| 23 | \( 1 + 2.40T + 23T^{2} \) |
| 29 | \( 1 - 0.267T + 29T^{2} \) |
| 31 | \( 1 - 1.40T + 31T^{2} \) |
| 37 | \( 1 + 5.46T + 37T^{2} \) |
| 43 | \( 1 + 3.21T + 43T^{2} \) |
| 47 | \( 1 - 0.201T + 47T^{2} \) |
| 53 | \( 1 + 6.42T + 53T^{2} \) |
| 59 | \( 1 - 6.58T + 59T^{2} \) |
| 61 | \( 1 - 9.30T + 61T^{2} \) |
| 67 | \( 1 + 13.5T + 67T^{2} \) |
| 71 | \( 1 - 12.4T + 71T^{2} \) |
| 73 | \( 1 + 13.5T + 73T^{2} \) |
| 79 | \( 1 + 15.4T + 79T^{2} \) |
| 83 | \( 1 + 4.93T + 83T^{2} \) |
| 89 | \( 1 + 9.68T + 89T^{2} \) |
| 97 | \( 1 + 5.14T + 97T^{2} \) |
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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
−7.38867132447040566713700792562, −6.76265132749429603037560497865, −6.24824381906193989002183628615, −5.44393933247540659442422569502, −4.58990839721759049728018774320, −4.05992686277558345732898012993, −3.69288757735890463732912205041, −2.79959759128697428490859855333, −1.48396897976259163642402310579, 0,
1.48396897976259163642402310579, 2.79959759128697428490859855333, 3.69288757735890463732912205041, 4.05992686277558345732898012993, 4.58990839721759049728018774320, 5.44393933247540659442422569502, 6.24824381906193989002183628615, 6.76265132749429603037560497865, 7.38867132447040566713700792562
