| L(s) = 1 | − 2-s + 1.09·3-s + 4-s − 1.58·5-s − 1.09·6-s + 3.71·7-s − 8-s − 1.81·9-s + 1.58·10-s − 0.650·11-s + 1.09·12-s + 3.57·13-s − 3.71·14-s − 1.73·15-s + 16-s − 6.23·17-s + 1.81·18-s + 19-s − 1.58·20-s + 4.05·21-s + 0.650·22-s + 1.26·23-s − 1.09·24-s − 2.48·25-s − 3.57·26-s − 5.24·27-s + 3.71·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.629·3-s + 0.5·4-s − 0.709·5-s − 0.445·6-s + 1.40·7-s − 0.353·8-s − 0.603·9-s + 0.501·10-s − 0.196·11-s + 0.314·12-s + 0.991·13-s − 0.993·14-s − 0.446·15-s + 0.250·16-s − 1.51·17-s + 0.426·18-s + 0.229·19-s − 0.354·20-s + 0.884·21-s + 0.138·22-s + 0.263·23-s − 0.222·24-s − 0.496·25-s − 0.700·26-s − 1.00·27-s + 0.702·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.706493499\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.706493499\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| 211 | \( 1 + T \) |
| good | 3 | \( 1 - 1.09T + 3T^{2} \) |
| 5 | \( 1 + 1.58T + 5T^{2} \) |
| 7 | \( 1 - 3.71T + 7T^{2} \) |
| 11 | \( 1 + 0.650T + 11T^{2} \) |
| 13 | \( 1 - 3.57T + 13T^{2} \) |
| 17 | \( 1 + 6.23T + 17T^{2} \) |
| 23 | \( 1 - 1.26T + 23T^{2} \) |
| 29 | \( 1 - 4.97T + 29T^{2} \) |
| 31 | \( 1 - 8.90T + 31T^{2} \) |
| 37 | \( 1 - 7.61T + 37T^{2} \) |
| 41 | \( 1 - 3.27T + 41T^{2} \) |
| 43 | \( 1 - 6.59T + 43T^{2} \) |
| 47 | \( 1 + 7.94T + 47T^{2} \) |
| 53 | \( 1 + 8.84T + 53T^{2} \) |
| 59 | \( 1 + 12.9T + 59T^{2} \) |
| 61 | \( 1 + 7.63T + 61T^{2} \) |
| 67 | \( 1 - 14.1T + 67T^{2} \) |
| 71 | \( 1 + 2.61T + 71T^{2} \) |
| 73 | \( 1 + 13.8T + 73T^{2} \) |
| 79 | \( 1 - 7.86T + 79T^{2} \) |
| 83 | \( 1 - 17.0T + 83T^{2} \) |
| 89 | \( 1 - 8.48T + 89T^{2} \) |
| 97 | \( 1 - 14.3T + 97T^{2} \) |
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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
−8.014823368713241090092547198459, −7.60199481972337079516950748873, −6.48232953921599312282325547415, −5.99950721021770252630352292717, −4.78263584101420727130680525397, −4.37712276515601369529750434940, −3.33199131186066008830208246089, −2.56333535583694017380314201171, −1.75537702474606477288282088022, −0.70413822409888664912676175351,
0.70413822409888664912676175351, 1.75537702474606477288282088022, 2.56333535583694017380314201171, 3.33199131186066008830208246089, 4.37712276515601369529750434940, 4.78263584101420727130680525397, 5.99950721021770252630352292717, 6.48232953921599312282325547415, 7.60199481972337079516950748873, 8.014823368713241090092547198459
