L(s) = 1 | − 1.73·7-s + i·13-s − 1.73i·19-s − 1.73i·31-s − 2i·37-s + 1.73·43-s + 1.99·49-s − 61-s − 1.73·67-s − 2i·73-s − 1.73i·91-s − i·97-s − 109-s + ⋯ |
L(s) = 1 | − 1.73·7-s + i·13-s − 1.73i·19-s − 1.73i·31-s − 2i·37-s + 1.73·43-s + 1.99·49-s − 61-s − 1.73·67-s − 2i·73-s − 1.73i·91-s − i·97-s − 109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0599 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0599 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7229596349\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7229596349\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + 1.73T + T^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 - iT - T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 + 1.73iT - T^{2} \) |
| 23 | \( 1 + T^{2} \) |
| 29 | \( 1 + T^{2} \) |
| 31 | \( 1 + 1.73iT - T^{2} \) |
| 37 | \( 1 + 2iT - T^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 - 1.73T + T^{2} \) |
| 47 | \( 1 + T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 + T + T^{2} \) |
| 67 | \( 1 + 1.73T + T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + 2iT - T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 + T^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 + iT - T^{2} \) |
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
−8.972812704191684147970736190894, −7.53651945173215191541791854077, −7.12128073383630226217343519377, −6.26041069058209161924812377322, −5.83001479492518291487328681494, −4.57375752149230854860579088637, −3.92124740640285264657476384108, −2.94743332839919575862384929869, −2.20882319214703421840765911854, −0.43772583544829272156730863230,
1.27362824469176788852176303353, 2.81111857194621833245509588558, 3.29369315525605633792016907440, 4.14176927804693017816704444295, 5.32054663692266614415694651963, 6.01313078680141796660719296579, 6.57247094323789106979156695380, 7.41210763121267428535238055622, 8.176836130389917512978189736210, 8.935221525037430308039466306781