L(s) = 1 | − 2-s − 3-s + 4-s + 5-s + 6-s − 7-s − 8-s + 9-s − 10-s − 1.60·11-s − 12-s − 1.10·13-s + 14-s − 15-s + 16-s + 0.493·17-s − 18-s − 2.49·19-s + 20-s + 21-s + 1.60·22-s + 23-s + 24-s + 25-s + 1.10·26-s − 27-s − 28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.447·5-s + 0.408·6-s − 0.377·7-s − 0.353·8-s + 0.333·9-s − 0.316·10-s − 0.483·11-s − 0.288·12-s − 0.307·13-s + 0.267·14-s − 0.258·15-s + 0.250·16-s + 0.119·17-s − 0.235·18-s − 0.572·19-s + 0.223·20-s + 0.218·21-s + 0.341·22-s + 0.208·23-s + 0.204·24-s + 0.200·25-s + 0.217·26-s − 0.192·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4830 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4830 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 - T \) |
good | 11 | \( 1 + 1.60T + 11T^{2} \) |
| 13 | \( 1 + 1.10T + 13T^{2} \) |
| 17 | \( 1 - 0.493T + 17T^{2} \) |
| 19 | \( 1 + 2.49T + 19T^{2} \) |
| 29 | \( 1 - 1.82T + 29T^{2} \) |
| 31 | \( 1 - 9.70T + 31T^{2} \) |
| 37 | \( 1 + 8.76T + 37T^{2} \) |
| 41 | \( 1 + 2.98T + 41T^{2} \) |
| 43 | \( 1 - 1.60T + 43T^{2} \) |
| 47 | \( 1 - 5.70T + 47T^{2} \) |
| 53 | \( 1 + 0.219T + 53T^{2} \) |
| 59 | \( 1 - 3.20T + 59T^{2} \) |
| 61 | \( 1 - 4.76T + 61T^{2} \) |
| 67 | \( 1 + 12.8T + 67T^{2} \) |
| 71 | \( 1 + 6.59T + 71T^{2} \) |
| 73 | \( 1 + 6T + 73T^{2} \) |
| 79 | \( 1 - 8.98T + 79T^{2} \) |
| 83 | \( 1 - 8.27T + 83T^{2} \) |
| 89 | \( 1 + 10.5T + 89T^{2} \) |
| 97 | \( 1 + 4.67T + 97T^{2} \) |
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
−7.980572519067874845775446823412, −7.08938273998004399694085377115, −6.58364250384176994163907298742, −5.84316348031716788200401526582, −5.13394340610657444213561324521, −4.26034908773327421447562299196, −3.08022625797751770350042256785, −2.28989875073263900648260160611, −1.18458113729292718591497223670, 0,
1.18458113729292718591497223670, 2.28989875073263900648260160611, 3.08022625797751770350042256785, 4.26034908773327421447562299196, 5.13394340610657444213561324521, 5.84316348031716788200401526582, 6.58364250384176994163907298742, 7.08938273998004399694085377115, 7.980572519067874845775446823412