L(s) = 1 | + (−0.488 + 0.488i)3-s + 4.71·7-s + 2.52i·9-s + (3.91 − 3.91i)11-s + (0.0878 − 0.0878i)13-s − 4.67i·17-s + (1.81 + 1.81i)19-s + (−2.30 + 2.30i)21-s + 1.63·23-s + (−2.69 − 2.69i)27-s + (−3.26 − 3.26i)29-s + 2.12·31-s + 3.82i·33-s + (−3.97 − 3.97i)37-s + 0.0858i·39-s + ⋯ |
L(s) = 1 | + (−0.282 + 0.282i)3-s + 1.78·7-s + 0.840i·9-s + (1.17 − 1.17i)11-s + (0.0243 − 0.0243i)13-s − 1.13i·17-s + (0.415 + 0.415i)19-s + (−0.502 + 0.502i)21-s + 0.339·23-s + (−0.519 − 0.519i)27-s + (−0.606 − 0.606i)29-s + 0.382·31-s + 0.665i·33-s + (−0.653 − 0.653i)37-s + 0.0137i·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.00745i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.00745i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.084169902\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.084169902\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
good | 3 | \( 1 + (0.488 - 0.488i)T - 3iT^{2} \) |
| 7 | \( 1 - 4.71T + 7T^{2} \) |
| 11 | \( 1 + (-3.91 + 3.91i)T - 11iT^{2} \) |
| 13 | \( 1 + (-0.0878 + 0.0878i)T - 13iT^{2} \) |
| 17 | \( 1 + 4.67iT - 17T^{2} \) |
| 19 | \( 1 + (-1.81 - 1.81i)T + 19iT^{2} \) |
| 23 | \( 1 - 1.63T + 23T^{2} \) |
| 29 | \( 1 + (3.26 + 3.26i)T + 29iT^{2} \) |
| 31 | \( 1 - 2.12T + 31T^{2} \) |
| 37 | \( 1 + (3.97 + 3.97i)T + 37iT^{2} \) |
| 41 | \( 1 - 8.25iT - 41T^{2} \) |
| 43 | \( 1 + (2.27 + 2.27i)T + 43iT^{2} \) |
| 47 | \( 1 + 4.06iT - 47T^{2} \) |
| 53 | \( 1 + (5.03 + 5.03i)T + 53iT^{2} \) |
| 59 | \( 1 + (5.16 - 5.16i)T - 59iT^{2} \) |
| 61 | \( 1 + (-7.12 - 7.12i)T + 61iT^{2} \) |
| 67 | \( 1 + (-7.49 + 7.49i)T - 67iT^{2} \) |
| 71 | \( 1 - 4.54iT - 71T^{2} \) |
| 73 | \( 1 + 8.30T + 73T^{2} \) |
| 79 | \( 1 - 11.5T + 79T^{2} \) |
| 83 | \( 1 + (1.16 - 1.16i)T - 83iT^{2} \) |
| 89 | \( 1 - 3.24iT - 89T^{2} \) |
| 97 | \( 1 - 13.9iT - 97T^{2} \) |
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
−9.332055285794210801105505390615, −8.522062584781126329895882133930, −7.931338770147489206947133782688, −7.13756265574311906915110690524, −5.95785968158158403022960664291, −5.19370106162561298058013174558, −4.57766805774227470137475965419, −3.54739586151840004201069956507, −2.16204972890977282422492531453, −1.06055400842787863970999252526,
1.26686859760065309716999775819, 1.89346727625711580446542061603, 3.59662479283307787782125896931, 4.46322341084658327538879884147, 5.19764969008805369693960910111, 6.26519276078795640323853859398, 7.02549071899323986076150541842, 7.73118161396279601390438998623, 8.675868195042873878531708218794, 9.252499259094909955651289114477