L(s) = 1 | + (1.87 − 1.08i)2-s + (−0.555 − 0.320i)3-s + (1.34 − 2.33i)4-s + (−2.93 + 1.69i)5-s − 1.39·6-s + (2.49 + 0.878i)7-s − 1.51i·8-s + (−1.29 − 2.24i)9-s + (−3.67 + 6.36i)10-s + (−3.01 + 1.37i)11-s + (−1.5 + 0.866i)12-s + 2.36·13-s + (5.63 − 1.05i)14-s + 2.17·15-s + (1.05 + 1.82i)16-s + (1.31 − 2.27i)17-s + ⋯ |
L(s) = 1 | + (1.32 − 0.766i)2-s + (−0.320 − 0.185i)3-s + (0.674 − 1.16i)4-s + (−1.31 + 0.758i)5-s − 0.567·6-s + (0.943 + 0.332i)7-s − 0.536i·8-s + (−0.431 − 0.747i)9-s + (−1.16 + 2.01i)10-s + (−0.910 + 0.414i)11-s + (−0.433 + 0.249i)12-s + 0.655·13-s + (1.50 − 0.281i)14-s + 0.562·15-s + (0.263 + 0.457i)16-s + (0.318 − 0.551i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.673 + 0.739i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.673 + 0.739i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.28159 - 0.565991i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.28159 - 0.565991i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 + (-2.49 - 0.878i)T \) |
| 11 | \( 1 + (3.01 - 1.37i)T \) |
good | 2 | \( 1 + (-1.87 + 1.08i)T + (1 - 1.73i)T^{2} \) |
| 3 | \( 1 + (0.555 + 0.320i)T + (1.5 + 2.59i)T^{2} \) |
| 5 | \( 1 + (2.93 - 1.69i)T + (2.5 - 4.33i)T^{2} \) |
| 13 | \( 1 - 2.36T + 13T^{2} \) |
| 17 | \( 1 + (-1.31 + 2.27i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (2.70 + 4.68i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (0.705 + 1.22i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 4.96iT - 29T^{2} \) |
| 31 | \( 1 + (2.54 + 1.47i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (0.699 + 1.21i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 7.09T + 41T^{2} \) |
| 43 | \( 1 + 4.74iT - 43T^{2} \) |
| 47 | \( 1 + (4.60 - 2.65i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-3.73 + 6.46i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (0.610 + 0.352i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-6.11 - 10.5i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (7.98 - 13.8i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 3.81T + 71T^{2} \) |
| 73 | \( 1 + (0.618 - 1.07i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-2.08 + 1.20i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 12.3T + 83T^{2} \) |
| 89 | \( 1 + (2.33 - 1.35i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 1.92iT - 97T^{2} \) |
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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
−14.45835940717029372144518919215, −13.03829177937034917371166277429, −12.05211455939353005575849056280, −11.34498075915107098512615281721, −10.76474261413311347431900119917, −8.492941276868035800165391025401, −7.10696430514525892154261782038, −5.49498531047969978366069168098, −4.19091141006818555497730561055, −2.82505853722408420752543010132,
3.85392829481407896944005942509, 4.82450831265332937774688670081, 5.84751129521819174048164302026, 7.82929198739877680783339764922, 8.160006081745530168162697753175, 10.69083471071593035714223341902, 11.64005007220479860829675264065, 12.65712194965155527580791086415, 13.65168696417160442311033467984, 14.66492991486530774856073133622