| L(s) = 1 | + 2-s + 3-s + 4-s − 5-s + 6-s + 7-s + 8-s + 9-s − 10-s + 11-s + 12-s + 14-s − 15-s + 16-s + 17-s + 18-s + 6·19-s − 20-s + 21-s + 22-s + 23-s + 24-s − 4·25-s + 27-s + 28-s + 29-s − 30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s − 0.316·10-s + 0.301·11-s + 0.288·12-s + 0.267·14-s − 0.258·15-s + 1/4·16-s + 0.242·17-s + 0.235·18-s + 1.37·19-s − 0.223·20-s + 0.218·21-s + 0.213·22-s + 0.208·23-s + 0.204·24-s − 4/5·25-s + 0.192·27-s + 0.188·28-s + 0.185·29-s − 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 189618 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189618 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.818913693\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.818913693\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 5 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 - 7 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 + 9 T + p T^{2} \) |
| 61 | \( 1 + 5 T + p T^{2} \) |
| 67 | \( 1 + 7 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 3 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 18 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
| 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
−13.12580095732758, −12.64293344293744, −12.22297592266184, −11.76349962359558, −11.34178137762567, −10.93209928855072, −10.30703736795157, −9.795473579300946, −9.275331122849289, −8.913824054915220, −8.109331904702249, −7.829891685077586, −7.392936732050511, −6.937376636065264, −6.276796989344393, −5.691870234058174, −5.287331797902039, −4.607162451849519, −4.166781204026830, −3.703487894702521, −3.008427530644477, −2.784554249823276, −1.802429884559966, −1.421752651741632, −0.5783418774738207,
0.5783418774738207, 1.421752651741632, 1.802429884559966, 2.784554249823276, 3.008427530644477, 3.703487894702521, 4.166781204026830, 4.607162451849519, 5.287331797902039, 5.691870234058174, 6.276796989344393, 6.937376636065264, 7.392936732050511, 7.829891685077586, 8.109331904702249, 8.913824054915220, 9.275331122849289, 9.795473579300946, 10.30703736795157, 10.93209928855072, 11.34178137762567, 11.76349962359558, 12.22297592266184, 12.64293344293744, 13.12580095732758
