L(s) = 1 | − 0.216·2-s + 3-s − 1.95·4-s + 3.49·5-s − 0.216·6-s − 7-s + 0.856·8-s + 9-s − 0.756·10-s + 2.73·11-s − 1.95·12-s + 7.01·13-s + 0.216·14-s + 3.49·15-s + 3.72·16-s + 3.20·17-s − 0.216·18-s + 0.351·19-s − 6.81·20-s − 21-s − 0.592·22-s + 4.72·23-s + 0.856·24-s + 7.18·25-s − 1.52·26-s + 27-s + 1.95·28-s + ⋯ |
L(s) = 1 | − 0.153·2-s + 0.577·3-s − 0.976·4-s + 1.56·5-s − 0.0884·6-s − 0.377·7-s + 0.302·8-s + 0.333·9-s − 0.239·10-s + 0.823·11-s − 0.563·12-s + 1.94·13-s + 0.0579·14-s + 0.901·15-s + 0.930·16-s + 0.776·17-s − 0.0510·18-s + 0.0806·19-s − 1.52·20-s − 0.218·21-s − 0.126·22-s + 0.984·23-s + 0.174·24-s + 1.43·25-s − 0.298·26-s + 0.192·27-s + 0.369·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.424181784\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.424181784\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 383 | \( 1 + T \) |
good | 2 | \( 1 + 0.216T + 2T^{2} \) |
| 5 | \( 1 - 3.49T + 5T^{2} \) |
| 11 | \( 1 - 2.73T + 11T^{2} \) |
| 13 | \( 1 - 7.01T + 13T^{2} \) |
| 17 | \( 1 - 3.20T + 17T^{2} \) |
| 19 | \( 1 - 0.351T + 19T^{2} \) |
| 23 | \( 1 - 4.72T + 23T^{2} \) |
| 29 | \( 1 - 9.38T + 29T^{2} \) |
| 31 | \( 1 + 3.60T + 31T^{2} \) |
| 37 | \( 1 + 3.25T + 37T^{2} \) |
| 41 | \( 1 - 3.87T + 41T^{2} \) |
| 43 | \( 1 - 3.82T + 43T^{2} \) |
| 47 | \( 1 + 6.68T + 47T^{2} \) |
| 53 | \( 1 + 1.64T + 53T^{2} \) |
| 59 | \( 1 + 1.34T + 59T^{2} \) |
| 61 | \( 1 - 12.6T + 61T^{2} \) |
| 67 | \( 1 + 6.40T + 67T^{2} \) |
| 71 | \( 1 + 10.3T + 71T^{2} \) |
| 73 | \( 1 + 5.81T + 73T^{2} \) |
| 79 | \( 1 + 6.10T + 79T^{2} \) |
| 83 | \( 1 - 6.59T + 83T^{2} \) |
| 89 | \( 1 + 7.09T + 89T^{2} \) |
| 97 | \( 1 + 5.57T + 97T^{2} \) |
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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
−8.140163696244355882755664074298, −7.04573274979094492934713847001, −6.34156922504662230406829372869, −5.81511925804688523216454752598, −5.11943676312163288677825210716, −4.18245722865673985176280251491, −3.47620283502161889110762521134, −2.77514866708788291311950081077, −1.42376175154036802657607432174, −1.13221820656353335314567708945,
1.13221820656353335314567708945, 1.42376175154036802657607432174, 2.77514866708788291311950081077, 3.47620283502161889110762521134, 4.18245722865673985176280251491, 5.11943676312163288677825210716, 5.81511925804688523216454752598, 6.34156922504662230406829372869, 7.04573274979094492934713847001, 8.140163696244355882755664074298