L(s) = 1 | − 2-s − 2.73·3-s + 4-s + 3.55·5-s + 2.73·6-s + 7-s − 8-s + 4.47·9-s − 3.55·10-s + 1.87·11-s − 2.73·12-s − 2.49·13-s − 14-s − 9.72·15-s + 16-s + 3.64·17-s − 4.47·18-s − 2.29·19-s + 3.55·20-s − 2.73·21-s − 1.87·22-s + 3.29·23-s + 2.73·24-s + 7.65·25-s + 2.49·26-s − 4.04·27-s + 28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1.57·3-s + 0.5·4-s + 1.59·5-s + 1.11·6-s + 0.377·7-s − 0.353·8-s + 1.49·9-s − 1.12·10-s + 0.566·11-s − 0.789·12-s − 0.691·13-s − 0.267·14-s − 2.51·15-s + 0.250·16-s + 0.883·17-s − 1.05·18-s − 0.525·19-s + 0.795·20-s − 0.596·21-s − 0.400·22-s + 0.686·23-s + 0.558·24-s + 1.53·25-s + 0.489·26-s − 0.778·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 431 | \( 1 - T \) |
good | 3 | \( 1 + 2.73T + 3T^{2} \) |
| 5 | \( 1 - 3.55T + 5T^{2} \) |
| 11 | \( 1 - 1.87T + 11T^{2} \) |
| 13 | \( 1 + 2.49T + 13T^{2} \) |
| 17 | \( 1 - 3.64T + 17T^{2} \) |
| 19 | \( 1 + 2.29T + 19T^{2} \) |
| 23 | \( 1 - 3.29T + 23T^{2} \) |
| 29 | \( 1 + 3.63T + 29T^{2} \) |
| 31 | \( 1 + 6.48T + 31T^{2} \) |
| 37 | \( 1 + 9.17T + 37T^{2} \) |
| 41 | \( 1 + 1.26T + 41T^{2} \) |
| 43 | \( 1 + 4.96T + 43T^{2} \) |
| 47 | \( 1 - 3.93T + 47T^{2} \) |
| 53 | \( 1 - 7.65T + 53T^{2} \) |
| 59 | \( 1 - 2.09T + 59T^{2} \) |
| 61 | \( 1 + 6.35T + 61T^{2} \) |
| 67 | \( 1 + 15.9T + 67T^{2} \) |
| 71 | \( 1 + 12.7T + 71T^{2} \) |
| 73 | \( 1 + 13.5T + 73T^{2} \) |
| 79 | \( 1 + 14.1T + 79T^{2} \) |
| 83 | \( 1 + 4.68T + 83T^{2} \) |
| 89 | \( 1 + 10.8T + 89T^{2} \) |
| 97 | \( 1 - 10.8T + 97T^{2} \) |
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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
−7.35069009299673128446141535001, −7.03575320459284791091186810186, −6.19076873073951768134345499869, −5.61407007721134344452155668821, −5.28012929712392818796453596708, −4.33556328595756940268440083445, −2.98333678623465459640297916746, −1.78112774623408782674305205532, −1.34819867416438774154270393488, 0,
1.34819867416438774154270393488, 1.78112774623408782674305205532, 2.98333678623465459640297916746, 4.33556328595756940268440083445, 5.28012929712392818796453596708, 5.61407007721134344452155668821, 6.19076873073951768134345499869, 7.03575320459284791091186810186, 7.35069009299673128446141535001