L(s) = 1 | + (−0.707 − 0.707i)5-s − 2i·13-s − 1.41·17-s + 1.00i·25-s − 1.41i·29-s − 1.41i·41-s + 49-s + 1.41·53-s − 2·61-s + (−1.41 + 1.41i)65-s − 2i·73-s + (1.00 + 1.00i)85-s + 1.41i·89-s + 2i·97-s + 1.41i·101-s + ⋯ |
L(s) = 1 | + (−0.707 − 0.707i)5-s − 2i·13-s − 1.41·17-s + 1.00i·25-s − 1.41i·29-s − 1.41i·41-s + 49-s + 1.41·53-s − 2·61-s + (−1.41 + 1.41i)65-s − 2i·73-s + (1.00 + 1.00i)85-s + 1.41i·89-s + 2i·97-s + 1.41i·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.169 + 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.169 + 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7674301355\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7674301355\) |
\(L(1)\) |
|
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 | \( 1 + (0.707 + 0.707i)T \) |
good | 7 | \( 1 - T^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 + 2iT - T^{2} \) |
| 17 | \( 1 + 1.41T + T^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( 1 + T^{2} \) |
| 29 | \( 1 + 1.41iT - T^{2} \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 + 1.41iT - T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + T^{2} \) |
| 53 | \( 1 - 1.41T + T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 + 2T + T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + 2iT - T^{2} \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( 1 + T^{2} \) |
| 89 | \( 1 - 1.41iT - T^{2} \) |
| 97 | \( 1 - 2iT - T^{2} \) |
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
−9.348997356311096425505065108385, −8.641718453927909592263013396562, −7.932405047998646393353733847485, −7.30476391923373773354794643674, −6.10737132164266983510691258898, −5.30183951511876939799916108282, −4.41983372176674333045357554816, −3.52914921997263042770272775758, −2.35102325481535181550146216901, −0.62134370471729405049938759768,
1.84536657162617498568943779418, 2.96049967869702040466268664607, 4.12683028261062857923536465982, 4.61925390597054316659859111273, 6.06310372854863886046778034149, 6.91140905814811315556539473835, 7.22874385489516996086569058046, 8.515600871157067927999478252506, 8.994557651964444412992615570431, 9.953044184929803010568313884903