L(s) = 1 | + (−0.5 + 0.866i)2-s + (−0.499 − 0.866i)4-s + (−0.5 − 0.866i)5-s + (0.5 − 0.866i)7-s + 0.999·8-s + 0.999·10-s + (0.5 − 0.866i)11-s + (0.499 + 0.866i)14-s + (−0.5 + 0.866i)16-s − 2·17-s + 19-s + (−0.499 + 0.866i)20-s + (0.499 + 0.866i)22-s + (0.5 + 0.866i)23-s − 0.999·28-s + ⋯ |
L(s) = 1 | + (−0.5 + 0.866i)2-s + (−0.499 − 0.866i)4-s + (−0.5 − 0.866i)5-s + (0.5 − 0.866i)7-s + 0.999·8-s + 0.999·10-s + (0.5 − 0.866i)11-s + (0.499 + 0.866i)14-s + (−0.5 + 0.866i)16-s − 2·17-s + 19-s + (−0.499 + 0.866i)20-s + (0.499 + 0.866i)22-s + (0.5 + 0.866i)23-s − 0.999·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.342 + 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.342 + 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6983292621\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6983292621\) |
\(L(1)\) |
|
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 | \( 1 \) |
| 7 | \( 1 + (-0.5 + 0.866i)T \) |
good | 5 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 17 | \( 1 + 2T + T^{2} \) |
| 19 | \( 1 - T + T^{2} \) |
| 23 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 31 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + T + T^{2} \) |
| 41 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 43 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 47 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 + T + T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 89 | \( 1 - T + T^{2} \) |
| 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
−8.804816752549850898130288785367, −8.426778567277041829403347678834, −7.46314202071736085099869323404, −6.98831923034405337640474694243, −5.99987358125071482286251452212, −5.08253181733135584145480783251, −4.43616968648803763528943977591, −3.62932963067065786443865299221, −1.72010137694663409963575000526, −0.59074092560479883092033320852,
1.67755590148562703536005320934, 2.54528131044469447678325367096, 3.38144261975425496257365794392, 4.44218798714335602936550527273, 5.08224948295402109156648650376, 6.59549761355052160381798194968, 7.09678196668526935685427506810, 7.952754639453290614692009359621, 8.911380546109075490693015780756, 9.144276481330495505466488822654