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
−9.920183222497986927222136320609, −9.264151279005790314926834886685, −8.749627743235234722843608667837, −7.32640268748989310065652447738, −6.65354347860048976690882664055, −5.62806545163682895828788526925, −5.08766156899489002738188140804, −4.19931125845241855403299471440, −2.78911155128203909008149399737, −1.81494789240979644213091350683,
0.17228390765535912651977531498, 1.70213945400167422386038251806, 3.11330832062020304508376558231, 3.71889080883961290304056420108, 5.29026924071801363371911525652, 6.16826543579900810778903628105, 6.40623479011797617720577058324, 7.64598478217002547092755861366, 8.072315191802798227350072544437, 9.492375023204944380056671799815