L(s) = 1 | − 2-s − 2.50·3-s + 4-s − 3.69·5-s + 2.50·6-s − 2.32·7-s − 8-s + 3.29·9-s + 3.69·10-s + 1.21·11-s − 2.50·12-s − 1.51·13-s + 2.32·14-s + 9.26·15-s + 16-s + 4.40·17-s − 3.29·18-s + 1.84·19-s − 3.69·20-s + 5.83·21-s − 1.21·22-s + 3.38·23-s + 2.50·24-s + 8.63·25-s + 1.51·26-s − 0.742·27-s − 2.32·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1.44·3-s + 0.5·4-s − 1.65·5-s + 1.02·6-s − 0.878·7-s − 0.353·8-s + 1.09·9-s + 1.16·10-s + 0.366·11-s − 0.724·12-s − 0.419·13-s + 0.621·14-s + 2.39·15-s + 0.250·16-s + 1.06·17-s − 0.776·18-s + 0.422·19-s − 0.825·20-s + 1.27·21-s − 0.258·22-s + 0.706·23-s + 0.512·24-s + 1.72·25-s + 0.296·26-s − 0.142·27-s − 0.439·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8002 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8002 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.3254148487\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3254148487\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 4001 | \( 1+O(T) \) |
good | 3 | \( 1 + 2.50T + 3T^{2} \) |
| 5 | \( 1 + 3.69T + 5T^{2} \) |
| 7 | \( 1 + 2.32T + 7T^{2} \) |
| 11 | \( 1 - 1.21T + 11T^{2} \) |
| 13 | \( 1 + 1.51T + 13T^{2} \) |
| 17 | \( 1 - 4.40T + 17T^{2} \) |
| 19 | \( 1 - 1.84T + 19T^{2} \) |
| 23 | \( 1 - 3.38T + 23T^{2} \) |
| 29 | \( 1 + 5.83T + 29T^{2} \) |
| 31 | \( 1 + 3.57T + 31T^{2} \) |
| 37 | \( 1 - 1.70T + 37T^{2} \) |
| 41 | \( 1 - 2.65T + 41T^{2} \) |
| 43 | \( 1 - 7.86T + 43T^{2} \) |
| 47 | \( 1 - 5.16T + 47T^{2} \) |
| 53 | \( 1 - 5.38T + 53T^{2} \) |
| 59 | \( 1 + 2.12T + 59T^{2} \) |
| 61 | \( 1 - 10.9T + 61T^{2} \) |
| 67 | \( 1 + 11.8T + 67T^{2} \) |
| 71 | \( 1 - 12.4T + 71T^{2} \) |
| 73 | \( 1 - 13.4T + 73T^{2} \) |
| 79 | \( 1 + 7.00T + 79T^{2} \) |
| 83 | \( 1 + 11.3T + 83T^{2} \) |
| 89 | \( 1 + 9.16T + 89T^{2} \) |
| 97 | \( 1 + 2.69T + 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.60601965371186328532669606773, −7.19644272521692624666185492399, −6.66092986739216333027554637856, −5.75342586593567968421340296274, −5.26199759291156437873913645973, −4.20998812623796742229298514691, −3.62071960862476914031453996684, −2.76144310457195681303054521897, −1.13596430382298707718110515390, −0.40934557332079275305356294915,
0.40934557332079275305356294915, 1.13596430382298707718110515390, 2.76144310457195681303054521897, 3.62071960862476914031453996684, 4.20998812623796742229298514691, 5.26199759291156437873913645973, 5.75342586593567968421340296274, 6.66092986739216333027554637856, 7.19644272521692624666185492399, 7.60601965371186328532669606773