| L(s) = 1 | − 7.37i·5-s + 1.26·7-s − 5.82i·11-s + 10.4·13-s + 18.3i·17-s + 20.8·19-s − 20.4i·23-s − 29.4·25-s − 11.1i·29-s + 61.3·31-s − 9.34i·35-s + 38.4·37-s − 33.0i·41-s + 49.3·43-s + 21.5i·47-s + ⋯ |
| L(s) = 1 | − 1.47i·5-s + 0.180·7-s − 0.529i·11-s + 0.806·13-s + 1.07i·17-s + 1.09·19-s − 0.890i·23-s − 1.17·25-s − 0.385i·29-s + 1.97·31-s − 0.266i·35-s + 1.03·37-s − 0.807i·41-s + 1.14·43-s + 0.459i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.213056883\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.213056883\) |
| \(L(2)\) |
|
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 | \( 1 + 7.37iT - 25T^{2} \) |
| 7 | \( 1 - 1.26T + 49T^{2} \) |
| 11 | \( 1 + 5.82iT - 121T^{2} \) |
| 13 | \( 1 - 10.4T + 169T^{2} \) |
| 17 | \( 1 - 18.3iT - 289T^{2} \) |
| 19 | \( 1 - 20.8T + 361T^{2} \) |
| 23 | \( 1 + 20.4iT - 529T^{2} \) |
| 29 | \( 1 + 11.1iT - 841T^{2} \) |
| 31 | \( 1 - 61.3T + 961T^{2} \) |
| 37 | \( 1 - 38.4T + 1.36e3T^{2} \) |
| 41 | \( 1 + 33.0iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 49.3T + 1.84e3T^{2} \) |
| 47 | \( 1 - 21.5iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 77.5iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 25.7iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 55.8T + 3.72e3T^{2} \) |
| 67 | \( 1 + 91.0T + 4.48e3T^{2} \) |
| 71 | \( 1 + 114. iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 120.T + 5.32e3T^{2} \) |
| 79 | \( 1 + 8.21T + 6.24e3T^{2} \) |
| 83 | \( 1 + 150. iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 118. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 72.8T + 9.40e3T^{2} \) |
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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
−8.789877124293276362711034536097, −8.311640705540671836434393680410, −7.63167436125054059526820064736, −6.23383703270615835117634527603, −5.78053926015889214845700990799, −4.69279587243954653746301367944, −4.13990131781773612160498897163, −2.91145048071967505700216232786, −1.45433132206426546669395842613, −0.70077206306656760896448132070,
1.17231258139626588283554867587, 2.60117490676669736400016079068, 3.19184949983705986024491780085, 4.27219451036568217959340501878, 5.34352541322195692620100489523, 6.26109380956074037465795980232, 7.00769717686139862543116558394, 7.56715005935015436582179736828, 8.440543589357136433587284435100, 9.665203229837728585474680109153
