| L(s) = 1 | − 4.89·5-s − 1.73·7-s + 8.48·11-s − 5.19i·13-s + 25.4i·17-s + 5i·19-s − 24.4i·23-s − 1.00·25-s + 39.1·29-s − 13.8·31-s + 8.48·35-s − 32.9i·37-s − 16.9i·41-s + 34i·43-s + 4.89i·47-s + ⋯ |
| L(s) = 1 | − 0.979·5-s − 0.247·7-s + 0.771·11-s − 0.399i·13-s + 1.49i·17-s + 0.263i·19-s − 1.06i·23-s − 0.0400·25-s + 1.35·29-s − 0.446·31-s + 0.242·35-s − 0.889i·37-s − 0.413i·41-s + 0.790i·43-s + 0.104i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.965 + 0.258i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.965 + 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.1445104147\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1445104147\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 4.89T + 25T^{2} \) |
| 7 | \( 1 + 1.73T + 49T^{2} \) |
| 11 | \( 1 - 8.48T + 121T^{2} \) |
| 13 | \( 1 + 5.19iT - 169T^{2} \) |
| 17 | \( 1 - 25.4iT - 289T^{2} \) |
| 19 | \( 1 - 5iT - 361T^{2} \) |
| 23 | \( 1 + 24.4iT - 529T^{2} \) |
| 29 | \( 1 - 39.1T + 841T^{2} \) |
| 31 | \( 1 + 13.8T + 961T^{2} \) |
| 37 | \( 1 + 32.9iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 16.9iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 34iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 4.89iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 9.79T + 2.80e3T^{2} \) |
| 59 | \( 1 - 76.3T + 3.48e3T^{2} \) |
| 61 | \( 1 + 22.5iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 19iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 68.5iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 25T + 5.32e3T^{2} \) |
| 79 | \( 1 + 140.T + 6.24e3T^{2} \) |
| 83 | \( 1 + 101.T + 6.88e3T^{2} \) |
| 89 | \( 1 - 8.48iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 71T + 9.40e3T^{2} \) |
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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
−8.426285279069833258084061481975, −8.268693613913160540275453237248, −7.15554887539442063850576846294, −6.43946496639640883103505386547, −5.60284582756363770973649287116, −4.30904069996316898632256637997, −3.87643710497339979112105862072, −2.80248196907302696212327813520, −1.39079489620171039806435681562, −0.04243087360422217423076410126,
1.24853302090461583480361544561, 2.75063029059348235117818650543, 3.64357463623479286001161519602, 4.46411777643149451341083761428, 5.32481799077827123663818723839, 6.52371187165779363442082373599, 7.10364862001093788792704073496, 7.86601957355220076786736474096, 8.738301683950905744318055424223, 9.443698018675842261623797203725
