| L(s) = 1 | − 0.477·2-s − 1.77·4-s + 5-s + 2.68·7-s + 1.80·8-s − 0.477·10-s − 4.66·13-s − 1.28·14-s + 2.68·16-s + 4.62·17-s − 4.34·19-s − 1.77·20-s − 2.77·23-s + 25-s + 2.22·26-s − 4.75·28-s + 3.01·29-s + 2.38·31-s − 4.88·32-s − 2.20·34-s + 2.68·35-s − 10.6·37-s + 2.07·38-s + 1.80·40-s + 2.21·41-s − 7.06·43-s + 1.32·46-s + ⋯ |
| L(s) = 1 | − 0.337·2-s − 0.886·4-s + 0.447·5-s + 1.01·7-s + 0.636·8-s − 0.150·10-s − 1.29·13-s − 0.342·14-s + 0.671·16-s + 1.12·17-s − 0.995·19-s − 0.396·20-s − 0.578·23-s + 0.200·25-s + 0.436·26-s − 0.899·28-s + 0.559·29-s + 0.428·31-s − 0.863·32-s − 0.378·34-s + 0.453·35-s − 1.74·37-s + 0.336·38-s + 0.284·40-s + 0.345·41-s − 1.07·43-s + 0.195·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5445 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5445 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 0.477T + 2T^{2} \) |
| 7 | \( 1 - 2.68T + 7T^{2} \) |
| 13 | \( 1 + 4.66T + 13T^{2} \) |
| 17 | \( 1 - 4.62T + 17T^{2} \) |
| 19 | \( 1 + 4.34T + 19T^{2} \) |
| 23 | \( 1 + 2.77T + 23T^{2} \) |
| 29 | \( 1 - 3.01T + 29T^{2} \) |
| 31 | \( 1 - 2.38T + 31T^{2} \) |
| 37 | \( 1 + 10.6T + 37T^{2} \) |
| 41 | \( 1 - 2.21T + 41T^{2} \) |
| 43 | \( 1 + 7.06T + 43T^{2} \) |
| 47 | \( 1 + 4.36T + 47T^{2} \) |
| 53 | \( 1 - 6.33T + 53T^{2} \) |
| 59 | \( 1 + 11.7T + 59T^{2} \) |
| 61 | \( 1 + 3.98T + 61T^{2} \) |
| 67 | \( 1 - 7.31T + 67T^{2} \) |
| 71 | \( 1 + 1.19T + 71T^{2} \) |
| 73 | \( 1 - 1.02T + 73T^{2} \) |
| 79 | \( 1 - 3.50T + 79T^{2} \) |
| 83 | \( 1 + 11.1T + 83T^{2} \) |
| 89 | \( 1 + 2.76T + 89T^{2} \) |
| 97 | \( 1 - 18.5T + 97T^{2} \) |
| 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
−7.972452238734435679336696737825, −7.32137368629280097587408176396, −6.38409801139034900413136551434, −5.38690133081401940733662626670, −4.94751070592654117607890514157, −4.30979494542898971879875734702, −3.28782635018257612269038357548, −2.13413600417048873456889724205, −1.33308199293790724793362069784, 0,
1.33308199293790724793362069784, 2.13413600417048873456889724205, 3.28782635018257612269038357548, 4.30979494542898971879875734702, 4.94751070592654117607890514157, 5.38690133081401940733662626670, 6.38409801139034900413136551434, 7.32137368629280097587408176396, 7.972452238734435679336696737825
