L(s) = 1 | + (−0.5 + 0.866i)2-s + (0.766 − 0.642i)3-s + (−0.499 − 0.866i)4-s + (0.173 + 0.984i)6-s + (−0.766 + 1.32i)7-s + 0.999·8-s + (0.173 − 0.984i)9-s + (−0.939 − 0.342i)12-s + (−0.173 − 0.300i)13-s + (−0.766 − 1.32i)14-s + (−0.5 + 0.866i)16-s + 1.53·17-s + (0.766 + 0.642i)18-s + 19-s + (0.266 + 1.50i)21-s + ⋯ |
L(s) = 1 | + (−0.5 + 0.866i)2-s + (0.766 − 0.642i)3-s + (−0.499 − 0.866i)4-s + (0.173 + 0.984i)6-s + (−0.766 + 1.32i)7-s + 0.999·8-s + (0.173 − 0.984i)9-s + (−0.939 − 0.342i)12-s + (−0.173 − 0.300i)13-s + (−0.766 − 1.32i)14-s + (−0.5 + 0.866i)16-s + 1.53·17-s + (0.766 + 0.642i)18-s + 19-s + (0.266 + 1.50i)21-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.018305143\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.018305143\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 - 0.866i)T \) |
| 3 | \( 1 + (-0.766 + 0.642i)T \) |
| 19 | \( 1 - T \) |
good | 5 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 7 | \( 1 + (0.766 - 1.32i)T + (-0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 + (0.173 + 0.300i)T + (-0.5 + 0.866i)T^{2} \) |
| 17 | \( 1 - 1.53T + T^{2} \) |
| 23 | \( 1 + (-0.939 - 1.62i)T + (-0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 + (-0.939 + 1.62i)T + (-0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + T + T^{2} \) |
| 41 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 43 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 47 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 - 0.347T + T^{2} \) |
| 59 | \( 1 + (0.173 + 0.300i)T + (-0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.939 - 1.62i)T + (-0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + 1.87T + T^{2} \) |
| 79 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + (0.5 + 0.866i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.591570408545238023479297107882, −9.037434650360248878680303657517, −8.177627876746804050256848198974, −7.52648608927184187366838138315, −6.82443111922298036261188674941, −5.68071136956470618976897104717, −5.45950954508005331637068321877, −3.64216651010247062419087711462, −2.73721145077242861032922701930, −1.34145304674740823708877979383,
1.17649345175262850142402949864, 2.81206684026711420083014862668, 3.39477090169108326329088829083, 4.25384542308664250855836331847, 5.09716343290955607216258659913, 6.80413679548074661144663545989, 7.48867137995250997605492501231, 8.273898121913476079985209389414, 9.127273726130011106257445560781, 9.799164529708777732269391546555