L(s) = 1 | + (−1 − 1.73i)2-s + 1.73i·3-s + (−1.99 + 3.46i)4-s − 2·5-s + (2.99 − 1.73i)6-s − 6.92i·7-s + 7.99·8-s − 2.99·9-s + (2 + 3.46i)10-s + 6.92i·11-s + (−5.99 − 3.46i)12-s + 2·13-s + (−11.9 + 6.92i)14-s − 3.46i·15-s + (−8 − 13.8i)16-s + 10·17-s + ⋯ |
L(s) = 1 | + (−0.5 − 0.866i)2-s + 0.577i·3-s + (−0.499 + 0.866i)4-s − 0.400·5-s + (0.499 − 0.288i)6-s − 0.989i·7-s + 0.999·8-s − 0.333·9-s + (0.200 + 0.346i)10-s + 0.629i·11-s + (−0.499 − 0.288i)12-s + 0.153·13-s + (−0.857 + 0.494i)14-s − 0.230i·15-s + (−0.5 − 0.866i)16-s + 0.588·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.866 + 0.5i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.866 + 0.5i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.541943 - 0.145213i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.541943 - 0.145213i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1 + 1.73i)T \) |
| 3 | \( 1 - 1.73iT \) |
good | 5 | \( 1 + 2T + 25T^{2} \) |
| 7 | \( 1 + 6.92iT - 49T^{2} \) |
| 11 | \( 1 - 6.92iT - 121T^{2} \) |
| 13 | \( 1 - 2T + 169T^{2} \) |
| 17 | \( 1 - 10T + 289T^{2} \) |
| 19 | \( 1 - 20.7iT - 361T^{2} \) |
| 23 | \( 1 + 27.7iT - 529T^{2} \) |
| 29 | \( 1 + 26T + 841T^{2} \) |
| 31 | \( 1 - 6.92iT - 961T^{2} \) |
| 37 | \( 1 - 26T + 1.36e3T^{2} \) |
| 41 | \( 1 - 58T + 1.68e3T^{2} \) |
| 43 | \( 1 + 48.4iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 69.2iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 74T + 2.80e3T^{2} \) |
| 59 | \( 1 + 90.0iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 26T + 3.72e3T^{2} \) |
| 67 | \( 1 - 6.92iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 5.04e3T^{2} \) |
| 73 | \( 1 + 46T + 5.32e3T^{2} \) |
| 79 | \( 1 - 117. iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 48.4iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 82T + 7.92e3T^{2} \) |
| 97 | \( 1 - 2T + 9.40e3T^{2} \) |
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
−20.21368534244247546625260313645, −18.88346699411507880536372215953, −17.33981884772548023069555570954, −16.24609613826692652410721547212, −14.27703873788863352896282392074, −12.51798678506732196418498026374, −10.93409908939634533731498037515, −9.751538032754624187969349089619, −7.82005355956976890465359257184, −4.04688556322157586076116458372,
5.81330292801615233284569326855, 7.71002150047828499977504048565, 9.174383214369402603361117159153, 11.45876558534976765996277661143, 13.35787087310893432453946414938, 14.93792911843039893260472154080, 16.10483807261337693382380385118, 17.62690934090527446578915444830, 18.74904406917276109914630132479, 19.63454363885461930623749185310