L(s) = 1 | − 2.51·2-s − 1.72·3-s + 4.30·4-s − 3.07·5-s + 4.34·6-s − 4.07·7-s − 5.79·8-s − 0.0123·9-s + 7.71·10-s + 11-s − 7.44·12-s + 0.957·13-s + 10.2·14-s + 5.31·15-s + 5.94·16-s + 17-s + 0.0311·18-s − 5.04·19-s − 13.2·20-s + 7.03·21-s − 2.51·22-s + 1.39·23-s + 10.0·24-s + 4.44·25-s − 2.40·26-s + 5.20·27-s − 17.5·28-s + ⋯ |
L(s) = 1 | − 1.77·2-s − 0.997·3-s + 2.15·4-s − 1.37·5-s + 1.77·6-s − 1.53·7-s − 2.04·8-s − 0.00413·9-s + 2.44·10-s + 0.301·11-s − 2.14·12-s + 0.265·13-s + 2.73·14-s + 1.37·15-s + 1.48·16-s + 0.242·17-s + 0.00733·18-s − 1.15·19-s − 2.95·20-s + 1.53·21-s − 0.535·22-s + 0.290·23-s + 2.04·24-s + 0.888·25-s − 0.471·26-s + 1.00·27-s − 3.31·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8041 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8041 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 11 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| 43 | \( 1 + T \) |
good | 2 | \( 1 + 2.51T + 2T^{2} \) |
| 3 | \( 1 + 1.72T + 3T^{2} \) |
| 5 | \( 1 + 3.07T + 5T^{2} \) |
| 7 | \( 1 + 4.07T + 7T^{2} \) |
| 13 | \( 1 - 0.957T + 13T^{2} \) |
| 19 | \( 1 + 5.04T + 19T^{2} \) |
| 23 | \( 1 - 1.39T + 23T^{2} \) |
| 29 | \( 1 + 8.87T + 29T^{2} \) |
| 31 | \( 1 + 0.871T + 31T^{2} \) |
| 37 | \( 1 + 3.42T + 37T^{2} \) |
| 41 | \( 1 + 4.61T + 41T^{2} \) |
| 47 | \( 1 + 10.9T + 47T^{2} \) |
| 53 | \( 1 - 7.25T + 53T^{2} \) |
| 59 | \( 1 + 9.48T + 59T^{2} \) |
| 61 | \( 1 + 6.63T + 61T^{2} \) |
| 67 | \( 1 + 7.78T + 67T^{2} \) |
| 71 | \( 1 + 14.6T + 71T^{2} \) |
| 73 | \( 1 - 15.3T + 73T^{2} \) |
| 79 | \( 1 - 14.9T + 79T^{2} \) |
| 83 | \( 1 + 7.35T + 83T^{2} \) |
| 89 | \( 1 - 6.13T + 89T^{2} \) |
| 97 | \( 1 + 14.7T + 97T^{2} \) |
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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
−7.47410527006347504181211230689, −6.96575558764678127014773656060, −6.35340609343460052697311988244, −5.87714880800210085050917148191, −4.67624903730089491705637612899, −3.61290926645960570188481784839, −3.09935538281183946417590927633, −1.82060945846565938792581070256, −0.54103209103695018031698565389, 0,
0.54103209103695018031698565389, 1.82060945846565938792581070256, 3.09935538281183946417590927633, 3.61290926645960570188481784839, 4.67624903730089491705637612899, 5.87714880800210085050917148191, 6.35340609343460052697311988244, 6.96575558764678127014773656060, 7.47410527006347504181211230689