L(s) = 1 | + (1.58 − 1.58i)5-s + (−3.23 − 3.23i)7-s − 4.57i·11-s + (−4.23 + 4.23i)13-s + (1.74 − 1.74i)17-s − 2.47i·19-s + (2.82 + 2.82i)23-s − 5.00i·25-s + 5.99·29-s + 1.52·31-s − 10.2·35-s + (2.23 + 2.23i)37-s − 7.07i·41-s + (2.47 − 2.47i)43-s + (−1.74 + 1.74i)47-s + ⋯ |
L(s) = 1 | + (0.707 − 0.707i)5-s + (−1.22 − 1.22i)7-s − 1.37i·11-s + (−1.17 + 1.17i)13-s + (0.423 − 0.423i)17-s − 0.567i·19-s + (0.589 + 0.589i)23-s − 1.00i·25-s + 1.11·29-s + 0.274·31-s − 1.72·35-s + (0.367 + 0.367i)37-s − 1.10i·41-s + (0.376 − 0.376i)43-s + (−0.254 + 0.254i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0618 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0618 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.796680 - 0.847591i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.796680 - 0.847591i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-1.58 + 1.58i)T \) |
good | 7 | \( 1 + (3.23 + 3.23i)T + 7iT^{2} \) |
| 11 | \( 1 + 4.57iT - 11T^{2} \) |
| 13 | \( 1 + (4.23 - 4.23i)T - 13iT^{2} \) |
| 17 | \( 1 + (-1.74 + 1.74i)T - 17iT^{2} \) |
| 19 | \( 1 + 2.47iT - 19T^{2} \) |
| 23 | \( 1 + (-2.82 - 2.82i)T + 23iT^{2} \) |
| 29 | \( 1 - 5.99T + 29T^{2} \) |
| 31 | \( 1 - 1.52T + 31T^{2} \) |
| 37 | \( 1 + (-2.23 - 2.23i)T + 37iT^{2} \) |
| 41 | \( 1 + 7.07iT - 41T^{2} \) |
| 43 | \( 1 + (-2.47 + 2.47i)T - 43iT^{2} \) |
| 47 | \( 1 + (1.74 - 1.74i)T - 47iT^{2} \) |
| 53 | \( 1 + 53iT^{2} \) |
| 59 | \( 1 - 1.08T + 59T^{2} \) |
| 61 | \( 1 - 10.4T + 61T^{2} \) |
| 67 | \( 1 + (1.52 + 1.52i)T + 67iT^{2} \) |
| 71 | \( 1 - 12.6iT - 71T^{2} \) |
| 73 | \( 1 + (0.527 - 0.527i)T - 73iT^{2} \) |
| 79 | \( 1 - 14.4iT - 79T^{2} \) |
| 83 | \( 1 + (-1.08 - 1.08i)T + 83iT^{2} \) |
| 89 | \( 1 - 0.746T + 89T^{2} \) |
| 97 | \( 1 + (-1 - i)T + 97iT^{2} \) |
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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
−11.15993933843407661520071790005, −10.04246955225837172549462345993, −9.521895035556398647843320202295, −8.601085493089828240476944051781, −7.20433124287065987305798816113, −6.48926497799519694047581823726, −5.30405227627593869601132994577, −4.12758134964691048001470966984, −2.81504521716507621262939527475, −0.793328701528245584417477695339,
2.33379078133200493294832038044, 3.10086776286565307561107331628, 4.96266267944122275292917979590, 5.97170564025349804095343286354, 6.76493149300548628641919907680, 7.84649566468525447246973160436, 9.196790076038281305302103272540, 9.975452548995130587311736026490, 10.33350984726477638000187493985, 11.93586441804593990388983897204