| L(s) = 1 | − 0.980·2-s − 3-s − 1.03·4-s + 1.00·5-s + 0.980·6-s − 3.53·7-s + 2.97·8-s + 9-s − 0.985·10-s + 6.32·11-s + 1.03·12-s + 3.86·13-s + 3.46·14-s − 1.00·15-s − 0.846·16-s − 17-s − 0.980·18-s − 2.59·19-s − 1.04·20-s + 3.53·21-s − 6.20·22-s + 6.45·23-s − 2.97·24-s − 3.99·25-s − 3.79·26-s − 27-s + 3.66·28-s + ⋯ |
| L(s) = 1 | − 0.693·2-s − 0.577·3-s − 0.518·4-s + 0.449·5-s + 0.400·6-s − 1.33·7-s + 1.05·8-s + 0.333·9-s − 0.311·10-s + 1.90·11-s + 0.299·12-s + 1.07·13-s + 0.925·14-s − 0.259·15-s − 0.211·16-s − 0.242·17-s − 0.231·18-s − 0.595·19-s − 0.233·20-s + 0.770·21-s − 1.32·22-s + 1.34·23-s − 0.608·24-s − 0.798·25-s − 0.743·26-s − 0.192·27-s + 0.692·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| 157 | \( 1 + T \) |
| good | 2 | \( 1 + 0.980T + 2T^{2} \) |
| 5 | \( 1 - 1.00T + 5T^{2} \) |
| 7 | \( 1 + 3.53T + 7T^{2} \) |
| 11 | \( 1 - 6.32T + 11T^{2} \) |
| 13 | \( 1 - 3.86T + 13T^{2} \) |
| 19 | \( 1 + 2.59T + 19T^{2} \) |
| 23 | \( 1 - 6.45T + 23T^{2} \) |
| 29 | \( 1 + 7.53T + 29T^{2} \) |
| 31 | \( 1 + 4.09T + 31T^{2} \) |
| 37 | \( 1 - 6.50T + 37T^{2} \) |
| 41 | \( 1 + 9.61T + 41T^{2} \) |
| 43 | \( 1 - 7.33T + 43T^{2} \) |
| 47 | \( 1 + 4.71T + 47T^{2} \) |
| 53 | \( 1 - 2.16T + 53T^{2} \) |
| 59 | \( 1 - 3.64T + 59T^{2} \) |
| 61 | \( 1 - 7.27T + 61T^{2} \) |
| 67 | \( 1 + 6.61T + 67T^{2} \) |
| 71 | \( 1 + 11.6T + 71T^{2} \) |
| 73 | \( 1 + 10.8T + 73T^{2} \) |
| 79 | \( 1 + 16.2T + 79T^{2} \) |
| 83 | \( 1 + 8.05T + 83T^{2} \) |
| 89 | \( 1 - 2.16T + 89T^{2} \) |
| 97 | \( 1 - 3.94T + 97T^{2} \) |
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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
−7.21831367470660528635650406546, −6.92772607562575534266973523894, −5.98948784784810188092814775855, −5.80518329661298004166481172116, −4.51803364576448047662310594436, −3.92404613644397706405324207769, −3.30700127716509272619982167035, −1.79937962502836335318712379453, −1.09545576732183928983972940713, 0,
1.09545576732183928983972940713, 1.79937962502836335318712379453, 3.30700127716509272619982167035, 3.92404613644397706405324207769, 4.51803364576448047662310594436, 5.80518329661298004166481172116, 5.98948784784810188092814775855, 6.92772607562575534266973523894, 7.21831367470660528635650406546
