| L(s) = 1 | + i·3-s + 1.32i·7-s − 9-s + 3.70·11-s + 3.32i·13-s − 1.61i·17-s + 19-s − 1.32·21-s − 7.67i·23-s − i·27-s + 1.70·29-s + 9.67·31-s + 3.70i·33-s − 5.96i·37-s − 3.32·39-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s + 0.499i·7-s − 0.333·9-s + 1.11·11-s + 0.921i·13-s − 0.391i·17-s + 0.229·19-s − 0.288·21-s − 1.59i·23-s − 0.192i·27-s + 0.317·29-s + 1.73·31-s + 0.645i·33-s − 0.980i·37-s − 0.531·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.177239105\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.177239105\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 7 | \( 1 - 1.32iT - 7T^{2} \) |
| 11 | \( 1 - 3.70T + 11T^{2} \) |
| 13 | \( 1 - 3.32iT - 13T^{2} \) |
| 17 | \( 1 + 1.61iT - 17T^{2} \) |
| 23 | \( 1 + 7.67iT - 23T^{2} \) |
| 29 | \( 1 - 1.70T + 29T^{2} \) |
| 31 | \( 1 - 9.67T + 31T^{2} \) |
| 37 | \( 1 + 5.96iT - 37T^{2} \) |
| 41 | \( 1 - 2.29T + 41T^{2} \) |
| 43 | \( 1 + 8.73iT - 43T^{2} \) |
| 47 | \( 1 + 11.6iT - 47T^{2} \) |
| 53 | \( 1 - 10.4iT - 53T^{2} \) |
| 59 | \( 1 + 12.6T + 59T^{2} \) |
| 61 | \( 1 - 9.02T + 61T^{2} \) |
| 67 | \( 1 - 13.2iT - 67T^{2} \) |
| 71 | \( 1 + 8.69T + 71T^{2} \) |
| 73 | \( 1 + 12.0iT - 73T^{2} \) |
| 79 | \( 1 - 14.0T + 79T^{2} \) |
| 83 | \( 1 - 5.02iT - 83T^{2} \) |
| 89 | \( 1 - 1.70T + 89T^{2} \) |
| 97 | \( 1 - 0.679iT - 97T^{2} \) |
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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
−8.476192325201717610354627727230, −7.38980373381824606086812377631, −6.63630332980254664377171969910, −6.12635940414968292572003713929, −5.21435787735443783987217441021, −4.40107009822396689234856969813, −3.94240091929509544741526529652, −2.84316729182076293004943236226, −2.08037912943365585739863573039, −0.76075255572067327470633421339,
0.901808010059099602905627644111, 1.52634412193335121002692261512, 2.82982319275267651540892595669, 3.50286962593426809745361721518, 4.40065823799091783049251158972, 5.21842245196070492395064885478, 6.23418283533634665271581736275, 6.46930684272339134340238132536, 7.54131827310886538843714037158, 7.87604051811680708517698812783
