L(s) = 1 | + (−1.58 + 1.58i)2-s + (−1 + i)3-s − 3.00i·4-s + (−2 + i)5-s − 3.16i·6-s + (1.58 + 1.58i)8-s + i·9-s + (1.58 − 4.74i)10-s + (1 + 3.16i)11-s + (3.00 + 3.00i)12-s + (3.16 + 3.16i)13-s + (1 − 3i)15-s + 0.999·16-s + (3.16 − 3.16i)17-s + (−1.58 − 1.58i)18-s − 6.32·19-s + ⋯ |
L(s) = 1 | + (−1.11 + 1.11i)2-s + (−0.577 + 0.577i)3-s − 1.50i·4-s + (−0.894 + 0.447i)5-s − 1.29i·6-s + (0.559 + 0.559i)8-s + 0.333i·9-s + (0.500 − 1.50i)10-s + (0.301 + 0.953i)11-s + (0.866 + 0.866i)12-s + (0.877 + 0.877i)13-s + (0.258 − 0.774i)15-s + 0.249·16-s + (0.766 − 0.766i)17-s + (−0.372 − 0.372i)18-s − 1.45·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.969 - 0.244i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.969 - 0.244i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0433953 + 0.349167i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0433953 + 0.349167i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (2 - i)T \) |
| 11 | \( 1 + (-1 - 3.16i)T \) |
good | 2 | \( 1 + (1.58 - 1.58i)T - 2iT^{2} \) |
| 3 | \( 1 + (1 - i)T - 3iT^{2} \) |
| 7 | \( 1 - 7iT^{2} \) |
| 13 | \( 1 + (-3.16 - 3.16i)T + 13iT^{2} \) |
| 17 | \( 1 + (-3.16 + 3.16i)T - 17iT^{2} \) |
| 19 | \( 1 + 6.32T + 19T^{2} \) |
| 23 | \( 1 + (1 - i)T - 23iT^{2} \) |
| 29 | \( 1 - 6.32T + 29T^{2} \) |
| 31 | \( 1 - 2T + 31T^{2} \) |
| 37 | \( 1 + (-3 - 3i)T + 37iT^{2} \) |
| 41 | \( 1 + 6.32iT - 41T^{2} \) |
| 43 | \( 1 + 43iT^{2} \) |
| 47 | \( 1 + (-3 - 3i)T + 47iT^{2} \) |
| 53 | \( 1 + (1 - i)T - 53iT^{2} \) |
| 59 | \( 1 + 6iT - 59T^{2} \) |
| 61 | \( 1 - 6.32iT - 61T^{2} \) |
| 67 | \( 1 + (-3 - 3i)T + 67iT^{2} \) |
| 71 | \( 1 + 8T + 71T^{2} \) |
| 73 | \( 1 + (3.16 + 3.16i)T + 73iT^{2} \) |
| 79 | \( 1 - 6.32T + 79T^{2} \) |
| 83 | \( 1 + (-6.32 - 6.32i)T + 83iT^{2} \) |
| 89 | \( 1 + 6iT - 89T^{2} \) |
| 97 | \( 1 + (7 + 7i)T + 97iT^{2} \) |
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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
−15.99415855402495081020900748480, −15.30452929556817541051919978519, −14.18921101149278700278016069382, −12.11982667470711951314687919039, −10.89944174647849326019021614739, −9.898819583401376212631158583627, −8.555156740161672823066773541136, −7.38974029877214882236176071903, −6.27960050060641700670605386254, −4.43129273790414915534528716769,
0.872296453380042686988010080295, 3.56769015484041802071715761194, 6.13754429632765401532898454305, 8.062227137583947885893163577620, 8.743308677963307139727859113879, 10.43076605168957514655098384351, 11.32660465527116186695832321215, 12.21118616335762390786025032360, 12.98485823157848293591959835471, 14.97974079313680884606138468962