| L(s) = 1 | + (−1.19 + 0.762i)2-s + (0.837 − 1.81i)4-s + i·5-s + 4.47i·7-s + (0.387 + 2.80i)8-s + (−0.762 − 1.19i)10-s − 0.776·11-s + 5.60·13-s + (−3.41 − 5.33i)14-s + (−2.59 − 3.04i)16-s − 3.09i·17-s + 8.33i·19-s + (1.81 + 0.837i)20-s + (0.924 − 0.591i)22-s − 4.30·23-s + ⋯ |
| L(s) = 1 | + (−0.842 + 0.539i)2-s + (0.418 − 0.908i)4-s + 0.447i·5-s + 1.69i·7-s + (0.136 + 0.990i)8-s + (−0.241 − 0.376i)10-s − 0.234·11-s + 1.55·13-s + (−0.912 − 1.42i)14-s + (−0.649 − 0.760i)16-s − 0.751i·17-s + 1.91i·19-s + (0.406 + 0.187i)20-s + (0.197 − 0.126i)22-s − 0.896·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.908 - 0.418i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.908 - 0.418i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9656379749\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9656379749\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.19 - 0.762i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - iT \) |
| good | 7 | \( 1 - 4.47iT - 7T^{2} \) |
| 11 | \( 1 + 0.776T + 11T^{2} \) |
| 13 | \( 1 - 5.60T + 13T^{2} \) |
| 17 | \( 1 + 3.09iT - 17T^{2} \) |
| 19 | \( 1 - 8.33iT - 19T^{2} \) |
| 23 | \( 1 + 4.30T + 23T^{2} \) |
| 29 | \( 1 - 10.3iT - 29T^{2} \) |
| 31 | \( 1 + 0.497iT - 31T^{2} \) |
| 37 | \( 1 - 7.15T + 37T^{2} \) |
| 41 | \( 1 + 8.67iT - 41T^{2} \) |
| 43 | \( 1 + 6.79iT - 43T^{2} \) |
| 47 | \( 1 - 4.18T + 47T^{2} \) |
| 53 | \( 1 - 3.17iT - 53T^{2} \) |
| 59 | \( 1 + 4.29T + 59T^{2} \) |
| 61 | \( 1 - 3.24T + 61T^{2} \) |
| 67 | \( 1 + 6.35iT - 67T^{2} \) |
| 71 | \( 1 + 6.89T + 71T^{2} \) |
| 73 | \( 1 + 2.87T + 73T^{2} \) |
| 79 | \( 1 - 2.81iT - 79T^{2} \) |
| 83 | \( 1 - 5.59T + 83T^{2} \) |
| 89 | \( 1 - 9.29iT - 89T^{2} \) |
| 97 | \( 1 + 3.12T + 97T^{2} \) |
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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
−9.504479400817115651260784237005, −8.840002351404908214718599668979, −8.295506700648374706450515089553, −7.50366360606332180746181003493, −6.41118091499202493986866454675, −5.83471326002531661661710335417, −5.30512402301365678791608547532, −3.68734040023510129736588330737, −2.52135950124522521212313983317, −1.53923730315415104390608861869,
0.53138827016581193622479940692, 1.41621226008870008210864199487, 2.84593380524144044986420960789, 4.04809436613541830768705729185, 4.37100956857647419960616386546, 6.09837379831897398332938335650, 6.76984838273024783980106938726, 7.83196023485880758326158330181, 8.147156547596403935495995578252, 9.149086320400394456451249831808
