L(s) = 1 | − i·2-s + (−0.618 + 1.61i)3-s − 4-s − 3.23·5-s + (1.61 + 0.618i)6-s + (2.61 + 0.381i)7-s + i·8-s + (−2.23 − 2.00i)9-s + 3.23i·10-s + (0.618 − 1.61i)12-s − i·13-s + (0.381 − 2.61i)14-s + (2.00 − 5.23i)15-s + 16-s − 2.47·17-s + (−2.00 + 2.23i)18-s + ⋯ |
L(s) = 1 | − 0.707i·2-s + (−0.356 + 0.934i)3-s − 0.5·4-s − 1.44·5-s + (0.660 + 0.252i)6-s + (0.989 + 0.144i)7-s + 0.353i·8-s + (−0.745 − 0.666i)9-s + 1.02i·10-s + (0.178 − 0.467i)12-s − 0.277i·13-s + (0.102 − 0.699i)14-s + (0.516 − 1.35i)15-s + 0.250·16-s − 0.599·17-s + (−0.471 + 0.527i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.487 + 0.872i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.487 + 0.872i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.283132 - 0.482645i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.283132 - 0.482645i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + iT \) |
| 3 | \( 1 + (0.618 - 1.61i)T \) |
| 7 | \( 1 + (-2.61 - 0.381i)T \) |
| 13 | \( 1 + iT \) |
good | 5 | \( 1 + 3.23T + 5T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 17 | \( 1 + 2.47T + 17T^{2} \) |
| 19 | \( 1 + 6iT - 19T^{2} \) |
| 23 | \( 1 + 3.23iT - 23T^{2} \) |
| 29 | \( 1 + 5.70iT - 29T^{2} \) |
| 31 | \( 1 + 7.23iT - 31T^{2} \) |
| 37 | \( 1 + 9.23T + 37T^{2} \) |
| 41 | \( 1 - 5.23T + 41T^{2} \) |
| 43 | \( 1 + 10.4T + 43T^{2} \) |
| 47 | \( 1 - 6.47T + 47T^{2} \) |
| 53 | \( 1 + 3.23iT - 53T^{2} \) |
| 59 | \( 1 - 4.47T + 59T^{2} \) |
| 61 | \( 1 + 4.47iT - 61T^{2} \) |
| 67 | \( 1 - 6.94T + 67T^{2} \) |
| 71 | \( 1 - 14.4iT - 71T^{2} \) |
| 73 | \( 1 - 3.23iT - 73T^{2} \) |
| 79 | \( 1 + 8.94T + 79T^{2} \) |
| 83 | \( 1 + 14T + 83T^{2} \) |
| 89 | \( 1 + 6.18T + 89T^{2} \) |
| 97 | \( 1 - 0.763iT - 97T^{2} \) |
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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
−10.82920534452968617411785385247, −9.807778948701095473042969965002, −8.693734720871506082321274230757, −8.216534980517118743093663847432, −6.98742680793422053909581370811, −5.43111265036818444640130116022, −4.49143929834660022589351200282, −3.96840321484916764839688827600, −2.63288593148908599925937862563, −0.35882198135387177808338923728,
1.50523886031027302027082048232, 3.54211697031364059506847117997, 4.67832086198914608700981358427, 5.59447061058866085654829214665, 6.91491668874169121820254979907, 7.40443081121605419777750268475, 8.257794059892876196197614131795, 8.719472560662486399619692099993, 10.47019959921805842096265426944, 11.26441287951958042426053598679