L(s) = 1 | − 2.45i·3-s + 1.48i·5-s − 3.05·9-s + 5.04i·11-s − 2.58i·13-s + 3.65·15-s + 1.25·17-s + 3.09i·19-s + 1.39·23-s + 2.79·25-s + 0.126i·27-s + 0.638i·29-s + 3.65·31-s + 12.4·33-s + 6.02i·37-s + ⋯ |
L(s) = 1 | − 1.42i·3-s + 0.664i·5-s − 1.01·9-s + 1.52i·11-s − 0.717i·13-s + 0.943·15-s + 0.305·17-s + 0.710i·19-s + 0.291·23-s + 0.558·25-s + 0.0243i·27-s + 0.118i·29-s + 0.656·31-s + 2.16·33-s + 0.989i·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.938 + 0.344i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.938 + 0.344i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.696323099\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.696323099\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + 2.45iT - 3T^{2} \) |
| 5 | \( 1 - 1.48iT - 5T^{2} \) |
| 11 | \( 1 - 5.04iT - 11T^{2} \) |
| 13 | \( 1 + 2.58iT - 13T^{2} \) |
| 17 | \( 1 - 1.25T + 17T^{2} \) |
| 19 | \( 1 - 3.09iT - 19T^{2} \) |
| 23 | \( 1 - 1.39T + 23T^{2} \) |
| 29 | \( 1 - 0.638iT - 29T^{2} \) |
| 31 | \( 1 - 3.65T + 31T^{2} \) |
| 37 | \( 1 - 6.02iT - 37T^{2} \) |
| 41 | \( 1 - 6.36T + 41T^{2} \) |
| 43 | \( 1 - 1.02iT - 43T^{2} \) |
| 47 | \( 1 - 10.9T + 47T^{2} \) |
| 53 | \( 1 - 5.76iT - 53T^{2} \) |
| 59 | \( 1 - 3.48iT - 59T^{2} \) |
| 61 | \( 1 + 12.8iT - 61T^{2} \) |
| 67 | \( 1 + 0.512iT - 67T^{2} \) |
| 71 | \( 1 - 7.41T + 71T^{2} \) |
| 73 | \( 1 - 9.89T + 73T^{2} \) |
| 79 | \( 1 + 8.70T + 79T^{2} \) |
| 83 | \( 1 - 2.97iT - 83T^{2} \) |
| 89 | \( 1 + 2.58T + 89T^{2} \) |
| 97 | \( 1 + 1.57T + 97T^{2} \) |
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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
−9.433302881152957034397102029811, −8.232397411366715785784896749959, −7.65655177660372090792790568751, −7.01808180555349954356708014534, −6.42930453986013798813899410915, −5.48388381524866650771802563903, −4.35529517276030355688629072902, −3.01540464702653400935230857081, −2.18591385701466262325883789833, −1.09329074943540572135412211094,
0.829402575728708751678381766355, 2.68467510947674333899622454794, 3.70265016527142071570211140206, 4.40812564271274973122998706572, 5.23818183498140904327994253981, 5.91914120158057155549930496577, 7.04182010534688018909918942681, 8.237196770136636773676231423690, 8.968527671793300274738851524132, 9.259520885232498960748371819648