L(s) = 1 | + (0.5 + 0.866i)2-s + (0.5 − 0.866i)3-s + (0.500 − 0.866i)4-s + (−1 − 1.73i)5-s + 0.999·6-s + 3·8-s + (−0.499 − 0.866i)9-s + (0.999 − 1.73i)10-s + (−2 + 3.46i)11-s + (−0.499 − 0.866i)12-s + 2·13-s − 1.99·15-s + (0.500 + 0.866i)16-s + (−3 + 5.19i)17-s + (0.499 − 0.866i)18-s + (2 + 3.46i)19-s + ⋯ |
L(s) = 1 | + (0.353 + 0.612i)2-s + (0.288 − 0.499i)3-s + (0.250 − 0.433i)4-s + (−0.447 − 0.774i)5-s + 0.408·6-s + 1.06·8-s + (−0.166 − 0.288i)9-s + (0.316 − 0.547i)10-s + (−0.603 + 1.04i)11-s + (−0.144 − 0.249i)12-s + 0.554·13-s − 0.516·15-s + (0.125 + 0.216i)16-s + (−0.727 + 1.26i)17-s + (0.117 − 0.204i)18-s + (0.458 + 0.794i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.968 + 0.250i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.968 + 0.250i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.43298 - 0.182610i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.43298 - 0.182610i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.5 + 0.866i)T \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (-0.5 - 0.866i)T + (-1 + 1.73i)T^{2} \) |
| 5 | \( 1 + (1 + 1.73i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (2 - 3.46i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 + (3 - 5.19i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-2 - 3.46i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (3 + 5.19i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (3 - 5.19i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-6 + 10.3i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (1 + 1.73i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (2 - 3.46i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + (3 - 5.19i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-8 - 13.8i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 12T + 83T^{2} \) |
| 89 | \( 1 + (7 + 12.1i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 18T + 97T^{2} \) |
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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
−13.02330501545202800489756566632, −12.37706777544349875402830591154, −11.03457210395954380294007920053, −9.950414922461296240777473019533, −8.525092380382120147497930696518, −7.66247519775145860938089500169, −6.56478569624224668215301513432, −5.37487154127442046284491509799, −4.13814467201104539154440020783, −1.77125131081290015271079329676,
2.75783616986689583278023535180, 3.53248899785197925579273264205, 4.98665985961532829464182856583, 6.79861725246347443155001270562, 7.84630499662072424196307054559, 8.986821541781670224412933437681, 10.46146611385686836717395235055, 11.19055537214842297399625531298, 11.75036984020560182464243259419, 13.35538967567012999742132355678