L(s) = 1 | − 1.47·2-s + 3-s + 0.169·4-s + 4.03·5-s − 1.47·6-s + 0.912·7-s + 2.69·8-s + 9-s − 5.94·10-s − 5.42·11-s + 0.169·12-s − 7.03·13-s − 1.34·14-s + 4.03·15-s − 4.30·16-s + 17-s − 1.47·18-s − 6.12·19-s + 0.683·20-s + 0.912·21-s + 7.98·22-s + 6.44·23-s + 2.69·24-s + 11.3·25-s + 10.3·26-s + 27-s + 0.154·28-s + ⋯ |
L(s) = 1 | − 1.04·2-s + 0.577·3-s + 0.0845·4-s + 1.80·5-s − 0.601·6-s + 0.344·7-s + 0.953·8-s + 0.333·9-s − 1.88·10-s − 1.63·11-s + 0.0488·12-s − 1.95·13-s − 0.359·14-s + 1.04·15-s − 1.07·16-s + 0.242·17-s − 0.347·18-s − 1.40·19-s + 0.152·20-s + 0.199·21-s + 1.70·22-s + 1.34·23-s + 0.550·24-s + 2.26·25-s + 2.03·26-s + 0.192·27-s + 0.0291·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| 79 | \( 1 + T \) |
good | 2 | \( 1 + 1.47T + 2T^{2} \) |
| 5 | \( 1 - 4.03T + 5T^{2} \) |
| 7 | \( 1 - 0.912T + 7T^{2} \) |
| 11 | \( 1 + 5.42T + 11T^{2} \) |
| 13 | \( 1 + 7.03T + 13T^{2} \) |
| 19 | \( 1 + 6.12T + 19T^{2} \) |
| 23 | \( 1 - 6.44T + 23T^{2} \) |
| 29 | \( 1 + 7.13T + 29T^{2} \) |
| 31 | \( 1 - 1.09T + 31T^{2} \) |
| 37 | \( 1 + 7.02T + 37T^{2} \) |
| 41 | \( 1 - 9.61T + 41T^{2} \) |
| 43 | \( 1 - 2.36T + 43T^{2} \) |
| 47 | \( 1 - 0.477T + 47T^{2} \) |
| 53 | \( 1 - 1.85T + 53T^{2} \) |
| 59 | \( 1 + 9.44T + 59T^{2} \) |
| 61 | \( 1 + 14.0T + 61T^{2} \) |
| 67 | \( 1 + 9.15T + 67T^{2} \) |
| 71 | \( 1 - 4.92T + 71T^{2} \) |
| 73 | \( 1 - 5.27T + 73T^{2} \) |
| 83 | \( 1 + 16.1T + 83T^{2} \) |
| 89 | \( 1 - 11.7T + 89T^{2} \) |
| 97 | \( 1 + 8.97T + 97T^{2} \) |
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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
−8.154662122220851525901025022807, −7.52119547826215541081375426013, −6.95688849864023850247508308776, −5.79232308246901457467663034124, −5.03961841991639299793466919099, −4.60188743128948441432522560758, −2.82182433394625979363566000759, −2.29098223124869524952549447627, −1.57143849246543699240768812523, 0,
1.57143849246543699240768812523, 2.29098223124869524952549447627, 2.82182433394625979363566000759, 4.60188743128948441432522560758, 5.03961841991639299793466919099, 5.79232308246901457467663034124, 6.95688849864023850247508308776, 7.52119547826215541081375426013, 8.154662122220851525901025022807