| L(s) = 1 | + (0.888 − 1.48i)3-s + (−1.29 + 1.82i)5-s − 0.470·7-s + (−1.42 − 2.64i)9-s − 2.71·11-s − 2.59i·13-s + (1.55 + 3.54i)15-s + 0.294i·17-s − 0.759i·19-s + (−0.417 + 0.699i)21-s + (−3.33 + 3.44i)23-s + (−1.62 − 4.72i)25-s + (−5.19 − 0.229i)27-s + 6.71i·29-s + 2.75·31-s + ⋯ |
| L(s) = 1 | + (0.512 − 0.858i)3-s + (−0.580 + 0.814i)5-s − 0.177·7-s + (−0.474 − 0.880i)9-s − 0.819·11-s − 0.719i·13-s + (0.401 + 0.915i)15-s + 0.0713i·17-s − 0.174i·19-s + (−0.0911 + 0.152i)21-s + (−0.694 + 0.719i)23-s + (−0.325 − 0.945i)25-s + (−0.999 − 0.0441i)27-s + 1.24i·29-s + 0.494·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.937 - 0.347i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.937 - 0.347i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1183013983\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1183013983\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.888 + 1.48i)T \) |
| 5 | \( 1 + (1.29 - 1.82i)T \) |
| 23 | \( 1 + (3.33 - 3.44i)T \) |
| good | 7 | \( 1 + 0.470T + 7T^{2} \) |
| 11 | \( 1 + 2.71T + 11T^{2} \) |
| 13 | \( 1 + 2.59iT - 13T^{2} \) |
| 17 | \( 1 - 0.294iT - 17T^{2} \) |
| 19 | \( 1 + 0.759iT - 19T^{2} \) |
| 29 | \( 1 - 6.71iT - 29T^{2} \) |
| 31 | \( 1 - 2.75T + 31T^{2} \) |
| 37 | \( 1 + 10.9T + 37T^{2} \) |
| 41 | \( 1 - 1.64iT - 41T^{2} \) |
| 43 | \( 1 + 3.08T + 43T^{2} \) |
| 47 | \( 1 + 9.53T + 47T^{2} \) |
| 53 | \( 1 + 2.10iT - 53T^{2} \) |
| 59 | \( 1 + 5.16iT - 59T^{2} \) |
| 61 | \( 1 - 8.99iT - 61T^{2} \) |
| 67 | \( 1 + 9.92T + 67T^{2} \) |
| 71 | \( 1 - 2.98iT - 71T^{2} \) |
| 73 | \( 1 + 8.50iT - 73T^{2} \) |
| 79 | \( 1 + 14.8iT - 79T^{2} \) |
| 83 | \( 1 + 10.6iT - 83T^{2} \) |
| 89 | \( 1 - 8.27T + 89T^{2} \) |
| 97 | \( 1 - 0.506T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.923015663073225062682778938360, −8.109618464764014449879628193041, −7.58200375886928986377307136556, −6.83590401046069612146510999397, −6.04559953466771379549484240680, −4.96109874996240633131618528899, −3.45255711838820428248705735094, −3.04633923008024844324659754976, −1.80289905691131733491718056381, −0.04174455542114843754309885987,
1.97738060306135855833248225601, 3.19367371216519189840496513946, 4.15088990653727809554757463324, 4.79244981627907003939569475034, 5.64316190823321706676127072529, 6.85495521571943760291378020748, 8.076647449756451371301353286028, 8.275595093178226402224761408842, 9.263651010436204018869342769274, 9.893808694721262843833779641880
