L(s) = 1 | + (−0.5 − 0.866i)3-s − 4-s + (0.5 + 0.866i)7-s + (−0.499 + 0.866i)9-s + (0.5 + 0.866i)12-s + (−1 − 1.73i)13-s + 16-s − 1.73i·19-s + (0.499 − 0.866i)21-s + (−0.5 − 0.866i)25-s + 0.999·27-s + (−0.5 − 0.866i)28-s − 2·31-s + (0.499 − 0.866i)36-s − 37-s + ⋯ |
L(s) = 1 | + (−0.5 − 0.866i)3-s − 4-s + (0.5 + 0.866i)7-s + (−0.499 + 0.866i)9-s + (0.5 + 0.866i)12-s + (−1 − 1.73i)13-s + 16-s − 1.73i·19-s + (0.499 − 0.866i)21-s + (−0.5 − 0.866i)25-s + 0.999·27-s + (−0.5 − 0.866i)28-s − 2·31-s + (0.499 − 0.866i)36-s − 37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1407 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.756 + 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1407 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.756 + 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4511849539\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4511849539\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.5 + 0.866i)T \) |
| 7 | \( 1 + (-0.5 - 0.866i)T \) |
| 67 | \( 1 + (-0.5 + 0.866i)T \) |
good | 2 | \( 1 + T^{2} \) |
| 5 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + T^{2} \) |
| 13 | \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \) |
| 17 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 19 | \( 1 + 1.73iT - T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 31 | \( 1 + 2T + T^{2} \) |
| 37 | \( 1 + T + T^{2} \) |
| 41 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 59 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 - T + T^{2} \) |
| 71 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 73 | \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \) |
| 79 | \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 89 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 97 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
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
−9.278152642972630635155646574095, −8.568451644253397834571473565804, −7.85050085512153876118821882790, −7.16971415059782852804438322047, −5.90873029274508255560536639429, −5.25998666943246839174554869352, −4.76349462022880674944190266461, −3.15141896464246794970969996187, −2.10852158794518297775379770277, −0.40780221694120514307641377259,
1.66084184773786321973719852678, 3.75925045005516498802534279134, 4.00244691971236529086078066226, 5.00088982328977428210566181828, 5.59799972855046629081492137786, 6.84320675288176762772422012347, 7.67086241007659697492344939252, 8.691575439596810060323304319486, 9.425090439393583766033409931193, 9.946617523557472932548620705314