L(s) = 1 | + 2-s + 1.16·3-s + 4-s − 5-s + 1.16·6-s − 7-s + 8-s − 1.63·9-s − 10-s + 1.16·12-s − 1.43·13-s − 14-s − 1.16·15-s + 16-s + 7.28·17-s − 1.63·18-s + 4.33·19-s − 20-s − 1.16·21-s + 2.02·23-s + 1.16·24-s + 25-s − 1.43·26-s − 5.41·27-s − 28-s − 9.11·29-s − 1.16·30-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.675·3-s + 0.5·4-s − 0.447·5-s + 0.477·6-s − 0.377·7-s + 0.353·8-s − 0.543·9-s − 0.316·10-s + 0.337·12-s − 0.397·13-s − 0.267·14-s − 0.302·15-s + 0.250·16-s + 1.76·17-s − 0.384·18-s + 0.994·19-s − 0.223·20-s − 0.255·21-s + 0.422·23-s + 0.238·24-s + 0.200·25-s − 0.281·26-s − 1.04·27-s − 0.188·28-s − 1.69·29-s − 0.213·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.601611596\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.601611596\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
good | 3 | \( 1 - 1.16T + 3T^{2} \) |
| 13 | \( 1 + 1.43T + 13T^{2} \) |
| 17 | \( 1 - 7.28T + 17T^{2} \) |
| 19 | \( 1 - 4.33T + 19T^{2} \) |
| 23 | \( 1 - 2.02T + 23T^{2} \) |
| 29 | \( 1 + 9.11T + 29T^{2} \) |
| 31 | \( 1 - 3.65T + 31T^{2} \) |
| 37 | \( 1 + 3.11T + 37T^{2} \) |
| 41 | \( 1 - 2.05T + 41T^{2} \) |
| 43 | \( 1 - 0.705T + 43T^{2} \) |
| 47 | \( 1 - 4.00T + 47T^{2} \) |
| 53 | \( 1 + 4.87T + 53T^{2} \) |
| 59 | \( 1 - 12.3T + 59T^{2} \) |
| 61 | \( 1 - 6.62T + 61T^{2} \) |
| 67 | \( 1 + 1.89T + 67T^{2} \) |
| 71 | \( 1 - 0.421T + 71T^{2} \) |
| 73 | \( 1 - 7.37T + 73T^{2} \) |
| 79 | \( 1 - 3.98T + 79T^{2} \) |
| 83 | \( 1 + 7.15T + 83T^{2} \) |
| 89 | \( 1 - 8.34T + 89T^{2} \) |
| 97 | \( 1 - 11.3T + 97T^{2} \) |
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
−7.57238436701965892522915601698, −7.35898952157588210379057685227, −6.33209447558201850984785323660, −5.49299156494887822423506556381, −5.18894779807275860266654517027, −4.00427349200044265507741558680, −3.41272037817581324742117171200, −2.96892400884381755039938853448, −2.04206739253327604492646005995, −0.798532337890934416239650085139,
0.798532337890934416239650085139, 2.04206739253327604492646005995, 2.96892400884381755039938853448, 3.41272037817581324742117171200, 4.00427349200044265507741558680, 5.18894779807275860266654517027, 5.49299156494887822423506556381, 6.33209447558201850984785323660, 7.35898952157588210379057685227, 7.57238436701965892522915601698