L(s) = 1 | + (0.5 − 0.866i)2-s + (−0.499 − 0.866i)4-s − i·7-s − 0.999·8-s + (−0.5 + 0.866i)9-s + (−0.866 − 1.5i)11-s + (−0.866 − 0.5i)14-s + (−0.5 + 0.866i)16-s + (0.499 + 0.866i)18-s − 1.73·22-s + (−0.5 − 0.866i)25-s + (−0.866 + 0.499i)28-s + (−0.866 − 1.5i)31-s + (0.499 + 0.866i)32-s + 0.999·36-s + (−0.5 + 0.866i)37-s + ⋯ |
L(s) = 1 | + (0.5 − 0.866i)2-s + (−0.499 − 0.866i)4-s − i·7-s − 0.999·8-s + (−0.5 + 0.866i)9-s + (−0.866 − 1.5i)11-s + (−0.866 − 0.5i)14-s + (−0.5 + 0.866i)16-s + (0.499 + 0.866i)18-s − 1.73·22-s + (−0.5 − 0.866i)25-s + (−0.866 + 0.499i)28-s + (−0.866 − 1.5i)31-s + (0.499 + 0.866i)32-s + 0.999·36-s + (−0.5 + 0.866i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2044 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 + 0.0633i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2044 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 + 0.0633i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9672079411\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9672079411\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 + 0.866i)T \) |
| 7 | \( 1 + iT \) |
| 73 | \( 1 + (0.5 + 0.866i)T \) |
good | 3 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 5 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 19 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 41 | \( 1 - T + T^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 59 | \( 1 + (-0.866 - 1.5i)T + (-0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 79 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + T^{2} \) |
| 89 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 97 | \( 1 - T + T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.995058225755487943241248056195, −8.208311600750909530973086667388, −7.56579569467137942712789811449, −6.25156989055431966382682895854, −5.63201295775174403501281275076, −4.79416406919865032826339211456, −3.88246306198453949217229640465, −3.03914992001668896080278639209, −2.10746567227129172603189829300, −0.55530706657110669692631933902,
2.16667374999513358154361896928, 3.14625876255394550909666513227, 4.11131941769775269611499841931, 5.25193305311067187187993552522, 5.52430144497344044638573151010, 6.60126987398262790096392481743, 7.24739832046858634768268204744, 8.029899934018987113597319868410, 8.947233756869108468647624907393, 9.327151527887792790001913187879