L(s) = 1 | + (−0.707 − 0.707i)3-s + (1.62 + 1.53i)5-s + (−2.42 − 2.42i)7-s + 1.00i·9-s − 4.29i·11-s + (−2.97 − 2.97i)13-s + (−0.0580 − 2.23i)15-s + (5.55 + 5.55i)17-s − 3.90·19-s + 3.42i·21-s + (−3.74 + 3.00i)23-s + (0.259 + 4.99i)25-s + (0.707 − 0.707i)27-s − 6.44i·29-s − 7.44·31-s + ⋯ |
L(s) = 1 | + (−0.408 − 0.408i)3-s + (0.725 + 0.688i)5-s + (−0.915 − 0.915i)7-s + 0.333i·9-s − 1.29i·11-s + (−0.825 − 0.825i)13-s + (−0.0149 − 0.577i)15-s + (1.34 + 1.34i)17-s − 0.896·19-s + 0.747i·21-s + (−0.780 + 0.625i)23-s + (0.0519 + 0.998i)25-s + (0.136 − 0.136i)27-s − 1.19i·29-s − 1.33·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.988 + 0.147i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.988 + 0.147i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4302592370\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4302592370\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.707 + 0.707i)T \) |
| 5 | \( 1 + (-1.62 - 1.53i)T \) |
| 23 | \( 1 + (3.74 - 3.00i)T \) |
good | 7 | \( 1 + (2.42 + 2.42i)T + 7iT^{2} \) |
| 11 | \( 1 + 4.29iT - 11T^{2} \) |
| 13 | \( 1 + (2.97 + 2.97i)T + 13iT^{2} \) |
| 17 | \( 1 + (-5.55 - 5.55i)T + 17iT^{2} \) |
| 19 | \( 1 + 3.90T + 19T^{2} \) |
| 29 | \( 1 + 6.44iT - 29T^{2} \) |
| 31 | \( 1 + 7.44T + 31T^{2} \) |
| 37 | \( 1 + (-1.45 - 1.45i)T + 37iT^{2} \) |
| 41 | \( 1 + 4.04T + 41T^{2} \) |
| 43 | \( 1 + (3.37 - 3.37i)T - 43iT^{2} \) |
| 47 | \( 1 + (7.41 - 7.41i)T - 47iT^{2} \) |
| 53 | \( 1 + (0.484 - 0.484i)T - 53iT^{2} \) |
| 59 | \( 1 + 15.1iT - 59T^{2} \) |
| 61 | \( 1 + 12.1iT - 61T^{2} \) |
| 67 | \( 1 + (-1.35 - 1.35i)T + 67iT^{2} \) |
| 71 | \( 1 + 14.1T + 71T^{2} \) |
| 73 | \( 1 + (8.24 + 8.24i)T + 73iT^{2} \) |
| 79 | \( 1 - 7.52T + 79T^{2} \) |
| 83 | \( 1 + (-3.22 + 3.22i)T - 83iT^{2} \) |
| 89 | \( 1 - 7.86T + 89T^{2} \) |
| 97 | \( 1 + (-4.32 - 4.32i)T + 97iT^{2} \) |
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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
−9.492375023204944380056671799815, −8.072315191802798227350072544437, −7.64598478217002547092755861366, −6.40623479011797617720577058324, −6.16826543579900810778903628105, −5.29026924071801363371911525652, −3.71889080883961290304056420108, −3.11330832062020304508376558231, −1.70213945400167422386038251806, −0.17228390765535912651977531498,
1.81494789240979644213091350683, 2.78911155128203909008149399737, 4.19931125845241855403299471440, 5.08766156899489002738188140804, 5.62806545163682895828788526925, 6.65354347860048976690882664055, 7.32640268748989310065652447738, 8.749627743235234722843608667837, 9.264151279005790314926834886685, 9.920183222497986927222136320609