L(s) = 1 | + (−1.36 − 1.36i)5-s + (−0.866 + 0.5i)9-s + (−0.866 − 0.5i)13-s + (−0.5 + 0.866i)17-s + 2.73i·25-s + (0.5 + 1.86i)29-s + (1.86 − 0.5i)37-s + (0.133 + 0.5i)41-s + (1.86 + 0.499i)45-s + (0.866 + 0.5i)49-s − 1.73i·53-s + (−0.133 + 0.5i)61-s + (0.499 + 1.86i)65-s + (0.366 + 0.366i)73-s + (0.499 − 0.866i)81-s + ⋯ |
L(s) = 1 | + (−1.36 − 1.36i)5-s + (−0.866 + 0.5i)9-s + (−0.866 − 0.5i)13-s + (−0.5 + 0.866i)17-s + 2.73i·25-s + (0.5 + 1.86i)29-s + (1.86 − 0.5i)37-s + (0.133 + 0.5i)41-s + (1.86 + 0.499i)45-s + (0.866 + 0.5i)49-s − 1.73i·53-s + (−0.133 + 0.5i)61-s + (0.499 + 1.86i)65-s + (0.366 + 0.366i)73-s + (0.499 − 0.866i)81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3536 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.362 - 0.931i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3536 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.362 - 0.931i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4741577468\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4741577468\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 13 | \( 1 + (0.866 + 0.5i)T \) |
| 17 | \( 1 + (0.5 - 0.866i)T \) |
good | 3 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 5 | \( 1 + (1.36 + 1.36i)T + iT^{2} \) |
| 7 | \( 1 + (-0.866 - 0.5i)T^{2} \) |
| 11 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 19 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
| 29 | \( 1 + (-0.5 - 1.86i)T + (-0.866 + 0.5i)T^{2} \) |
| 31 | \( 1 - iT^{2} \) |
| 37 | \( 1 + (-1.86 + 0.5i)T + (0.866 - 0.5i)T^{2} \) |
| 41 | \( 1 + (-0.133 - 0.5i)T + (-0.866 + 0.5i)T^{2} \) |
| 43 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 47 | \( 1 + T^{2} \) |
| 53 | \( 1 + 1.73iT - T^{2} \) |
| 59 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (0.133 - 0.5i)T + (-0.866 - 0.5i)T^{2} \) |
| 67 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 73 | \( 1 + (-0.366 - 0.366i)T + iT^{2} \) |
| 79 | \( 1 - iT^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + (1.73 + i)T + (0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 + (0.366 - 1.36i)T + (-0.866 - 0.5i)T^{2} \) |
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
−8.779991352907270427163022671993, −8.079967421403103569746542007677, −7.75700095153280611454008781710, −6.79598479456892938990318670266, −5.62668248278256712182863518850, −5.03585923563431173113643545123, −4.37846922207251840726286887221, −3.54429478534341812860920301305, −2.50930555492044522717238162855, −1.09270127979804506080579312899,
0.31685156408038657137136491756, 2.61985193593352546955837338700, 2.79915261356301970685975486834, 4.02687499181533259968519147528, 4.46863654289251538628080483949, 5.78372108708983851532078614276, 6.55299237641110208913208842483, 7.13518151012563174958806640850, 7.80236709710352661363560920436, 8.405124208327778116771676073300