L(s) = 1 | − 1.77i·5-s + (2.39 − 1.12i)7-s − 3.85·11-s − 6.59·13-s + 4.56·17-s + 1.05·19-s − 0.0946i·23-s + 1.83·25-s − 4.31·29-s − 4.07i·31-s + (−2.00 − 4.25i)35-s + 4.65i·37-s − 11.5·41-s + 6.28i·43-s − 6.12·47-s + ⋯ |
L(s) = 1 | − 0.794i·5-s + (0.904 − 0.426i)7-s − 1.16·11-s − 1.82·13-s + 1.10·17-s + 0.241·19-s − 0.0197i·23-s + 0.367·25-s − 0.801·29-s − 0.732i·31-s + (−0.339 − 0.718i)35-s + 0.764i·37-s − 1.80·41-s + 0.958i·43-s − 0.894·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.906 - 0.422i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.906 - 0.422i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.2167158540\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2167158540\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-2.39 + 1.12i)T \) |
good | 5 | \( 1 + 1.77iT - 5T^{2} \) |
| 11 | \( 1 + 3.85T + 11T^{2} \) |
| 13 | \( 1 + 6.59T + 13T^{2} \) |
| 17 | \( 1 - 4.56T + 17T^{2} \) |
| 19 | \( 1 - 1.05T + 19T^{2} \) |
| 23 | \( 1 + 0.0946iT - 23T^{2} \) |
| 29 | \( 1 + 4.31T + 29T^{2} \) |
| 31 | \( 1 + 4.07iT - 31T^{2} \) |
| 37 | \( 1 - 4.65iT - 37T^{2} \) |
| 41 | \( 1 + 11.5T + 41T^{2} \) |
| 43 | \( 1 - 6.28iT - 43T^{2} \) |
| 47 | \( 1 + 6.12T + 47T^{2} \) |
| 53 | \( 1 + 2.44T + 53T^{2} \) |
| 59 | \( 1 + 13.1iT - 59T^{2} \) |
| 61 | \( 1 - 4.58T + 61T^{2} \) |
| 67 | \( 1 - 4.83iT - 67T^{2} \) |
| 71 | \( 1 + 11.5iT - 71T^{2} \) |
| 73 | \( 1 - 2.25iT - 73T^{2} \) |
| 79 | \( 1 + 14.2T + 79T^{2} \) |
| 83 | \( 1 - 10.6iT - 83T^{2} \) |
| 89 | \( 1 + 8.61T + 89T^{2} \) |
| 97 | \( 1 - 14.2iT - 97T^{2} \) |
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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
−7.903530178699819954539031590510, −7.59163244228395952386326029962, −6.69110786236316771017766521313, −5.30230622993489908317924045488, −5.17149643081032655656135362916, −4.48677846821219157527963724381, −3.30571550773589697217013483964, −2.33007560275573520823516418695, −1.32191257393848222709548169356, −0.05906886222958664385675404714,
1.70418516931113654864884623413, 2.61886399469883831304715359919, 3.19517197283192641888328303615, 4.48435215181117000506138974714, 5.32622899127881045362806338678, 5.55955238041885707570122221534, 6.99539602642435355581824033965, 7.32315100988377265356671929402, 8.017651418606052995359744732026, 8.725877843951336811212374127913