L(s) = 1 | − 2.56·2-s − 3-s + 4.57·4-s − 4.02·5-s + 2.56·6-s + 7-s − 6.58·8-s + 9-s + 10.3·10-s − 2.04·11-s − 4.57·12-s − 3.98·13-s − 2.56·14-s + 4.02·15-s + 7.74·16-s − 2.64·17-s − 2.56·18-s + 0.955·19-s − 18.3·20-s − 21-s + 5.24·22-s + 5.68·23-s + 6.58·24-s + 11.1·25-s + 10.2·26-s − 27-s + 4.57·28-s + ⋯ |
L(s) = 1 | − 1.81·2-s − 0.577·3-s + 2.28·4-s − 1.79·5-s + 1.04·6-s + 0.377·7-s − 2.32·8-s + 0.333·9-s + 3.26·10-s − 0.616·11-s − 1.31·12-s − 1.10·13-s − 0.685·14-s + 1.03·15-s + 1.93·16-s − 0.641·17-s − 0.604·18-s + 0.219·19-s − 4.11·20-s − 0.218·21-s + 1.11·22-s + 1.18·23-s + 1.34·24-s + 2.23·25-s + 2.00·26-s − 0.192·27-s + 0.863·28-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
\(L(1)\) |
\(\approx\) |
\(0.1628204100\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1628204100\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.56T + 2T^{2} \) |
| 5 | \( 1 + 4.02T + 5T^{2} \) |
| 11 | \( 1 + 2.04T + 11T^{2} \) |
| 13 | \( 1 + 3.98T + 13T^{2} \) |
| 17 | \( 1 + 2.64T + 17T^{2} \) |
| 19 | \( 1 - 0.955T + 19T^{2} \) |
| 23 | \( 1 - 5.68T + 23T^{2} \) |
| 29 | \( 1 - 2.11T + 29T^{2} \) |
| 31 | \( 1 + 3.54T + 31T^{2} \) |
| 37 | \( 1 - 7.26T + 37T^{2} \) |
| 41 | \( 1 + 2.24T + 41T^{2} \) |
| 43 | \( 1 - 9.17T + 43T^{2} \) |
| 47 | \( 1 + 10.5T + 47T^{2} \) |
| 53 | \( 1 - 0.903T + 53T^{2} \) |
| 59 | \( 1 + 5.90T + 59T^{2} \) |
| 61 | \( 1 + 4.64T + 61T^{2} \) |
| 67 | \( 1 + 5.76T + 67T^{2} \) |
| 71 | \( 1 + 5.68T + 71T^{2} \) |
| 73 | \( 1 + 12.0T + 73T^{2} \) |
| 79 | \( 1 + 2.09T + 79T^{2} \) |
| 83 | \( 1 + 0.102T + 83T^{2} \) |
| 89 | \( 1 + 9.25T + 89T^{2} \) |
| 97 | \( 1 - 17.4T + 97T^{2} \) |
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
−7.64294914796553988922673595321, −7.51288005616690340740536279577, −6.96092511384003213489014073724, −6.07114258347983812806882683252, −4.90405951239549372929157094066, −4.45120029876720533681333090389, −3.20453929519691455414171023453, −2.48839570863090510275893213658, −1.25311509092516418432807559537, −0.30063529198402385849201926803,
0.30063529198402385849201926803, 1.25311509092516418432807559537, 2.48839570863090510275893213658, 3.20453929519691455414171023453, 4.45120029876720533681333090389, 4.90405951239549372929157094066, 6.07114258347983812806882683252, 6.96092511384003213489014073724, 7.51288005616690340740536279577, 7.64294914796553988922673595321