| L(s) = 1 | − 2.11·2-s + 2.47·4-s − 1.64·5-s − 1.00·8-s + 3.47·10-s − 11-s + 4.58·13-s − 2.83·16-s − 0.715·17-s + 1.64·19-s − 4.06·20-s + 2.11·22-s − 2.30·25-s − 9.70·26-s + 5.53·29-s − 5.17·31-s + 7.98·32-s + 1.51·34-s − 6.58·37-s − 3.47·38-s + 1.64·40-s − 5.17·41-s + 5.17·43-s − 2.47·44-s − 7.87·47-s + 4.87·50-s + 11.3·52-s + ⋯ |
| L(s) = 1 | − 1.49·2-s + 1.23·4-s − 0.734·5-s − 0.353·8-s + 1.09·10-s − 0.301·11-s + 1.27·13-s − 0.707·16-s − 0.173·17-s + 0.376·19-s − 0.907·20-s + 0.450·22-s − 0.460·25-s − 1.90·26-s + 1.02·29-s − 0.929·31-s + 1.41·32-s + 0.259·34-s − 1.08·37-s − 0.563·38-s + 0.259·40-s − 0.808·41-s + 0.789·43-s − 0.372·44-s − 1.14·47-s + 0.688·50-s + 1.57·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4851 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4851 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 2 | \( 1 + 2.11T + 2T^{2} \) |
| 5 | \( 1 + 1.64T + 5T^{2} \) |
| 13 | \( 1 - 4.58T + 13T^{2} \) |
| 17 | \( 1 + 0.715T + 17T^{2} \) |
| 19 | \( 1 - 1.64T + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 - 5.53T + 29T^{2} \) |
| 31 | \( 1 + 5.17T + 31T^{2} \) |
| 37 | \( 1 + 6.58T + 37T^{2} \) |
| 41 | \( 1 + 5.17T + 41T^{2} \) |
| 43 | \( 1 - 5.17T + 43T^{2} \) |
| 47 | \( 1 + 7.87T + 47T^{2} \) |
| 53 | \( 1 + 2.71T + 53T^{2} \) |
| 59 | \( 1 - 3.15T + 59T^{2} \) |
| 61 | \( 1 - 13.1T + 61T^{2} \) |
| 67 | \( 1 - 4.21T + 67T^{2} \) |
| 71 | \( 1 - 1.85T + 71T^{2} \) |
| 73 | \( 1 + 10.4T + 73T^{2} \) |
| 79 | \( 1 + 12.4T + 79T^{2} \) |
| 83 | \( 1 - 9.17T + 83T^{2} \) |
| 89 | \( 1 + 4.71T + 89T^{2} \) |
| 97 | \( 1 + 13.6T + 97T^{2} \) |
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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
−8.205072735492725954939448066088, −7.37237427911344212883776099998, −6.83384181965931783888485408761, −5.96972120620192306456042147375, −4.97932535564447771296592367505, −4.00338524071715987967253666346, −3.23367722312576722244846821763, −2.03367505357677935346513491876, −1.09473251633718558066328793532, 0,
1.09473251633718558066328793532, 2.03367505357677935346513491876, 3.23367722312576722244846821763, 4.00338524071715987967253666346, 4.97932535564447771296592367505, 5.96972120620192306456042147375, 6.83384181965931783888485408761, 7.37237427911344212883776099998, 8.205072735492725954939448066088
