| L(s) = 1 | + i·2-s − 4-s + (2 + i)5-s + 0.449i·7-s − i·8-s + (−1 + 2i)10-s + 4.89·11-s + 4i·13-s − 0.449·14-s + 16-s + 4.89i·17-s − 8.44·19-s + (−2 − i)20-s + 4.89i·22-s − 0.898i·23-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.5·4-s + (0.894 + 0.447i)5-s + 0.169i·7-s − 0.353i·8-s + (−0.316 + 0.632i)10-s + 1.47·11-s + 1.10i·13-s − 0.120·14-s + 0.250·16-s + 1.18i·17-s − 1.93·19-s + (−0.447 − 0.223i)20-s + 1.04i·22-s − 0.187i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3330 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3330 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.821975435\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.821975435\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 + (-2 - i)T \) |
| 37 | \( 1 + iT \) |
| good | 7 | \( 1 - 0.449iT - 7T^{2} \) |
| 11 | \( 1 - 4.89T + 11T^{2} \) |
| 13 | \( 1 - 4iT - 13T^{2} \) |
| 17 | \( 1 - 4.89iT - 17T^{2} \) |
| 19 | \( 1 + 8.44T + 19T^{2} \) |
| 23 | \( 1 + 0.898iT - 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + 6.44T + 31T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 + 0.449iT - 47T^{2} \) |
| 53 | \( 1 + 7.79iT - 53T^{2} \) |
| 59 | \( 1 + 8.44T + 59T^{2} \) |
| 61 | \( 1 - 12T + 61T^{2} \) |
| 67 | \( 1 - 10.4iT - 67T^{2} \) |
| 71 | \( 1 - 4.89T + 71T^{2} \) |
| 73 | \( 1 - 4iT - 73T^{2} \) |
| 79 | \( 1 + 1.55T + 79T^{2} \) |
| 83 | \( 1 - 14.4iT - 83T^{2} \) |
| 89 | \( 1 + 3.79T + 89T^{2} \) |
| 97 | \( 1 - 2iT - 97T^{2} \) |
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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
−8.850751727114278066216479546679, −8.409281281858559878727918982510, −7.15907211979321098484002732423, −6.50800644727922372061970308572, −6.27242369128473060173550432937, −5.36276285823082138273837300217, −4.21222382493353566844289108862, −3.78909877979420307068281951418, −2.25514342503771385329676757305, −1.52170477230154553816749219646,
0.54480829931739303927939302466, 1.61636034350377696011495375979, 2.47195283835074942003540811684, 3.53815455958854310181144654529, 4.37015708223829015969982634648, 5.16477346938447626857722608463, 5.98861725166168852055516137501, 6.68653493332165323383809837537, 7.64819645990110061954764837974, 8.724849945058854896472707960452
