| L(s) = 1 | + (−0.618 + 1.61i)3-s − 5-s + (−2.61 − 0.381i)7-s + (−2.23 − 2.00i)9-s − 5.23i·11-s + 3.23i·13-s + (0.618 − 1.61i)15-s − 4.47·17-s − 2.76i·19-s + (2.23 − 4i)21-s − 5.70i·23-s + 25-s + (4.61 − 2.38i)27-s + 4i·29-s − 1.23i·31-s + ⋯ |
| L(s) = 1 | + (−0.356 + 0.934i)3-s − 0.447·5-s + (−0.989 − 0.144i)7-s + (−0.745 − 0.666i)9-s − 1.57i·11-s + 0.897i·13-s + (0.159 − 0.417i)15-s − 1.08·17-s − 0.634i·19-s + (0.487 − 0.872i)21-s − 1.19i·23-s + 0.200·25-s + (0.888 − 0.458i)27-s + 0.742i·29-s − 0.222i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.487 + 0.872i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.487 + 0.872i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.124026 - 0.211422i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.124026 - 0.211422i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.618 - 1.61i)T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + (2.61 + 0.381i)T \) |
| good | 11 | \( 1 + 5.23iT - 11T^{2} \) |
| 13 | \( 1 - 3.23iT - 13T^{2} \) |
| 17 | \( 1 + 4.47T + 17T^{2} \) |
| 19 | \( 1 + 2.76iT - 19T^{2} \) |
| 23 | \( 1 + 5.70iT - 23T^{2} \) |
| 29 | \( 1 - 4iT - 29T^{2} \) |
| 31 | \( 1 + 1.23iT - 31T^{2} \) |
| 37 | \( 1 + 4.47T + 37T^{2} \) |
| 41 | \( 1 + 12.4T + 41T^{2} \) |
| 43 | \( 1 + 2.76T + 43T^{2} \) |
| 47 | \( 1 - 1.23T + 47T^{2} \) |
| 53 | \( 1 - 4.76iT - 53T^{2} \) |
| 59 | \( 1 + 8.94T + 59T^{2} \) |
| 61 | \( 1 + 12.9iT - 61T^{2} \) |
| 67 | \( 1 + 3.70T + 67T^{2} \) |
| 71 | \( 1 - 10.1iT - 71T^{2} \) |
| 73 | \( 1 - 11.2iT - 73T^{2} \) |
| 79 | \( 1 - 1.52T + 79T^{2} \) |
| 83 | \( 1 + 2.76T + 83T^{2} \) |
| 89 | \( 1 + 3.52T + 89T^{2} \) |
| 97 | \( 1 + 8.18iT - 97T^{2} \) |
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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
−10.95400822419770385531984116764, −10.10506408188090517335480710382, −8.945265584855927875779921625023, −8.615641671499098792217135958581, −6.83449330724216762560194347890, −6.23569567507222654965781937079, −4.94404154571157515864879251935, −3.88963108707000860014831668196, −2.97806278520627561433182190339, −0.15540025662594980281334859618,
1.93037446463870524463219336394, 3.33415260326266015183406669701, 4.82396254806836367321165365139, 5.97703583517688998697097113948, 6.94296996869852914715587024927, 7.56823755887703292829274846459, 8.638857161421144420072504192118, 9.793065740119662753219509063532, 10.58969996281387121727032836640, 11.82699699171567963645873581815
