| L(s) = 1 | + 1.73·3-s − 13.2·7-s + 2.99·9-s + 11.4i·11-s − i·13-s − 31.8i·17-s + 13.2i·19-s − 22.8·21-s + 32.2·23-s + 5.19·27-s + 4.10·29-s − 37.4i·31-s + 19.8i·33-s + 17.7i·37-s − 1.73i·39-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.88·7-s + 0.333·9-s + 1.04i·11-s − 0.0769i·13-s − 1.87i·17-s + 0.695i·19-s − 1.09·21-s + 1.40·23-s + 0.192·27-s + 0.141·29-s − 1.20i·31-s + 0.603i·33-s + 0.481i·37-s − 0.0444i·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.834 + 0.550i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.834 + 0.550i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.724932403\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.724932403\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 1.73T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 13.2T + 49T^{2} \) |
| 11 | \( 1 - 11.4iT - 121T^{2} \) |
| 13 | \( 1 + iT - 169T^{2} \) |
| 17 | \( 1 + 31.8iT - 289T^{2} \) |
| 19 | \( 1 - 13.2iT - 361T^{2} \) |
| 23 | \( 1 - 32.2T + 529T^{2} \) |
| 29 | \( 1 - 4.10T + 841T^{2} \) |
| 31 | \( 1 + 37.4iT - 961T^{2} \) |
| 37 | \( 1 - 17.7iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 53.6T + 1.68e3T^{2} \) |
| 43 | \( 1 - 24.8T + 1.84e3T^{2} \) |
| 47 | \( 1 - 35.5T + 2.20e3T^{2} \) |
| 53 | \( 1 + 89.6iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 6.00iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 76.7T + 3.72e3T^{2} \) |
| 67 | \( 1 - 108.T + 4.48e3T^{2} \) |
| 71 | \( 1 - 1.09iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 30.2iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 2.54iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 72.7T + 6.88e3T^{2} \) |
| 89 | \( 1 - 96T + 7.92e3T^{2} \) |
| 97 | \( 1 + 114. iT - 9.40e3T^{2} \) |
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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
−9.595935111657475958754628275786, −8.966266742985093720629798889162, −7.63864781339092016772358497221, −7.07493694250296666355683321542, −6.33272158444363857912815092547, −5.19449636162823422413253859540, −4.10098922595302323853720789913, −3.10877708945399068011746315529, −2.42187580578727596094434023801, −0.61293862245930293229243852477,
0.923063551153198503014540626477, 2.64654183006300153562233437330, 3.33659089668741558592167666573, 4.12778432973056606441949280341, 5.62465319589326997730823994518, 6.36878081792317208804822844902, 7.03346248922645269801445736726, 8.120471461182027544346333373033, 9.123744014442149111464235227917, 9.246123039902727706400358492193
