| L(s) = 1 | − 2.54·2-s − 2.75·3-s + 4.47·4-s − 0.0821·5-s + 7.00·6-s − 2.51·7-s − 6.29·8-s + 4.57·9-s + 0.209·10-s − 4.33·11-s − 12.3·12-s − 6.03·13-s + 6.39·14-s + 0.226·15-s + 7.07·16-s − 4.63·17-s − 11.6·18-s + 1.23·19-s − 0.367·20-s + 6.91·21-s + 11.0·22-s − 8.79·23-s + 17.3·24-s − 4.99·25-s + 15.3·26-s − 4.33·27-s − 11.2·28-s + ⋯ |
| L(s) = 1 | − 1.79·2-s − 1.58·3-s + 2.23·4-s − 0.0367·5-s + 2.85·6-s − 0.950·7-s − 2.22·8-s + 1.52·9-s + 0.0661·10-s − 1.30·11-s − 3.55·12-s − 1.67·13-s + 1.70·14-s + 0.0584·15-s + 1.76·16-s − 1.12·17-s − 2.74·18-s + 0.282·19-s − 0.0822·20-s + 1.50·21-s + 2.35·22-s − 1.83·23-s + 3.53·24-s − 0.998·25-s + 3.00·26-s − 0.833·27-s − 2.12·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 53 | \( 1 + T \) |
| 151 | \( 1 + T \) |
| good | 2 | \( 1 + 2.54T + 2T^{2} \) |
| 3 | \( 1 + 2.75T + 3T^{2} \) |
| 5 | \( 1 + 0.0821T + 5T^{2} \) |
| 7 | \( 1 + 2.51T + 7T^{2} \) |
| 11 | \( 1 + 4.33T + 11T^{2} \) |
| 13 | \( 1 + 6.03T + 13T^{2} \) |
| 17 | \( 1 + 4.63T + 17T^{2} \) |
| 19 | \( 1 - 1.23T + 19T^{2} \) |
| 23 | \( 1 + 8.79T + 23T^{2} \) |
| 29 | \( 1 + 3.61T + 29T^{2} \) |
| 31 | \( 1 + 6.33T + 31T^{2} \) |
| 37 | \( 1 - 2.98T + 37T^{2} \) |
| 41 | \( 1 - 11.0T + 41T^{2} \) |
| 43 | \( 1 + 0.747T + 43T^{2} \) |
| 47 | \( 1 - 4.74T + 47T^{2} \) |
| 59 | \( 1 - 15.0T + 59T^{2} \) |
| 61 | \( 1 + 8.06T + 61T^{2} \) |
| 67 | \( 1 - 1.53T + 67T^{2} \) |
| 71 | \( 1 - 10.6T + 71T^{2} \) |
| 73 | \( 1 - 0.140T + 73T^{2} \) |
| 79 | \( 1 - 9.74T + 79T^{2} \) |
| 83 | \( 1 - 7.93T + 83T^{2} \) |
| 89 | \( 1 + 1.05T + 89T^{2} \) |
| 97 | \( 1 - 8.11T + 97T^{2} \) |
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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
−7.58770578302217836981301961765, −6.95880259657413778703659879232, −6.24036402373392921526427370494, −5.73467283656161095615043567504, −4.97065982068757445900138847470, −3.92908637427478396839296184227, −2.48875258717126517188370260349, −2.07598449294396351634114984964, −0.49260617093514507323352169460, 0,
0.49260617093514507323352169460, 2.07598449294396351634114984964, 2.48875258717126517188370260349, 3.92908637427478396839296184227, 4.97065982068757445900138847470, 5.73467283656161095615043567504, 6.24036402373392921526427370494, 6.95880259657413778703659879232, 7.58770578302217836981301961765
