| L(s) = 1 | − 2.48·2-s − 3-s + 4.17·4-s − 2.72·5-s + 2.48·6-s + 7-s − 5.40·8-s + 9-s + 6.77·10-s + 1.40·11-s − 4.17·12-s − 3.18·13-s − 2.48·14-s + 2.72·15-s + 5.08·16-s + 6.94·17-s − 2.48·18-s − 5.84·19-s − 11.3·20-s − 21-s − 3.48·22-s + 2.50·23-s + 5.40·24-s + 2.43·25-s + 7.92·26-s − 27-s + 4.17·28-s + ⋯ |
| L(s) = 1 | − 1.75·2-s − 0.577·3-s + 2.08·4-s − 1.21·5-s + 1.01·6-s + 0.377·7-s − 1.91·8-s + 0.333·9-s + 2.14·10-s + 0.422·11-s − 1.20·12-s − 0.884·13-s − 0.664·14-s + 0.704·15-s + 1.27·16-s + 1.68·17-s − 0.585·18-s − 1.34·19-s − 2.54·20-s − 0.218·21-s − 0.742·22-s + 0.521·23-s + 1.10·24-s + 0.487·25-s + 1.55·26-s − 0.192·27-s + 0.789·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 383 | \( 1 - T \) |
| good | 2 | \( 1 + 2.48T + 2T^{2} \) |
| 5 | \( 1 + 2.72T + 5T^{2} \) |
| 11 | \( 1 - 1.40T + 11T^{2} \) |
| 13 | \( 1 + 3.18T + 13T^{2} \) |
| 17 | \( 1 - 6.94T + 17T^{2} \) |
| 19 | \( 1 + 5.84T + 19T^{2} \) |
| 23 | \( 1 - 2.50T + 23T^{2} \) |
| 29 | \( 1 - 4.45T + 29T^{2} \) |
| 31 | \( 1 - 1.94T + 31T^{2} \) |
| 37 | \( 1 + 8.67T + 37T^{2} \) |
| 41 | \( 1 - 1.44T + 41T^{2} \) |
| 43 | \( 1 + 6.61T + 43T^{2} \) |
| 47 | \( 1 - 3.78T + 47T^{2} \) |
| 53 | \( 1 + 11.8T + 53T^{2} \) |
| 59 | \( 1 - 3.46T + 59T^{2} \) |
| 61 | \( 1 - 6.46T + 61T^{2} \) |
| 67 | \( 1 - 12.5T + 67T^{2} \) |
| 71 | \( 1 + 1.58T + 71T^{2} \) |
| 73 | \( 1 - 14.7T + 73T^{2} \) |
| 79 | \( 1 + 15.0T + 79T^{2} \) |
| 83 | \( 1 - 1.14T + 83T^{2} \) |
| 89 | \( 1 + 12.1T + 89T^{2} \) |
| 97 | \( 1 - 0.826T + 97T^{2} \) |
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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
−7.66540686449612398287398585623, −6.97895753966473622141766614303, −6.60284420357966001801228276736, −5.50948568415251605888989867766, −4.70168538261703242478533047513, −3.82269973680804912523880141228, −2.87313714908159101013817670385, −1.78706958335993893977292540165, −0.888740340058306508479227674135, 0,
0.888740340058306508479227674135, 1.78706958335993893977292540165, 2.87313714908159101013817670385, 3.82269973680804912523880141228, 4.70168538261703242478533047513, 5.50948568415251605888989867766, 6.60284420357966001801228276736, 6.97895753966473622141766614303, 7.66540686449612398287398585623
