L(s) = 1 | + 1.69·2-s + 0.888·4-s + (−0.474 + 0.822i)5-s − 1.88·8-s + (−0.806 + 1.39i)10-s + (−0.294 − 0.509i)11-s + (2.50 + 4.34i)13-s − 4.98·16-s + (−3.79 + 6.56i)17-s + (2.23 + 3.86i)19-s + (−0.421 + 0.730i)20-s + (−0.5 − 0.866i)22-s + (1.23 − 2.14i)23-s + (2.04 + 3.54i)25-s + (4.26 + 7.38i)26-s + ⋯ |
L(s) = 1 | + 1.20·2-s + 0.444·4-s + (−0.212 + 0.367i)5-s − 0.667·8-s + (−0.255 + 0.441i)10-s + (−0.0886 − 0.153i)11-s + (0.696 + 1.20i)13-s − 1.24·16-s + (−0.919 + 1.59i)17-s + (0.511 + 0.886i)19-s + (−0.0943 + 0.163i)20-s + (−0.106 − 0.184i)22-s + (0.258 − 0.447i)23-s + (0.409 + 0.709i)25-s + (0.836 + 1.44i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0538 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0538 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.193997409\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.193997409\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 - 1.69T + 2T^{2} \) |
| 5 | \( 1 + (0.474 - 0.822i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (0.294 + 0.509i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-2.50 - 4.34i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (3.79 - 6.56i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.23 - 3.86i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.23 + 2.14i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-2.73 + 4.74i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 6.07T + 31T^{2} \) |
| 37 | \( 1 + (-3.49 - 6.05i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (0.527 + 0.913i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (3.49 - 6.05i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 - 7.47T + 47T^{2} \) |
| 53 | \( 1 + (-3.46 + 5.99i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + 10.4T + 59T^{2} \) |
| 61 | \( 1 - 11.6T + 61T^{2} \) |
| 67 | \( 1 + 11.8T + 67T^{2} \) |
| 71 | \( 1 + 4.30T + 71T^{2} \) |
| 73 | \( 1 + (2.23 - 3.86i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + 1.33T + 79T^{2} \) |
| 83 | \( 1 + (2.84 - 4.92i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-0.421 - 0.730i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-1.70 + 2.94i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−9.897338712768152180017525630662, −8.907543492513571493093718614782, −8.311432265235887330938814834354, −7.06519670374042906367128721506, −6.30497343997637880137507008198, −5.72805747268866245537802747738, −4.50336173874167921114045368950, −3.95656149219504402464756954029, −3.07908913615572750064050538736, −1.76719316640938997915984305179,
0.60556554721176308445564204384, 2.58131740208272035024608110006, 3.34570063312164063292503750511, 4.42421132870510771231274328289, 5.10653194103196350530087062352, 5.75582321078820903096886231542, 6.85204725616969424474764697661, 7.57395493495847382123586537325, 8.921838605535840192007054132990, 9.067788522514340865211610290847