L(s) = 1 | + (−0.5 + 0.866i)2-s + (−0.724 − 1.57i)3-s + (−0.499 − 0.866i)4-s + 1.44·5-s + (1.72 + 0.158i)6-s + 0.999·8-s + (−1.94 + 2.28i)9-s + (−0.724 + 1.25i)10-s + 2·11-s + (−1 + 1.41i)12-s + (2.44 − 4.24i)13-s + (−1.05 − 2.28i)15-s + (−0.5 + 0.866i)16-s + (1 − 1.73i)17-s + (−0.999 − 2.82i)18-s + (1.27 + 2.20i)19-s + ⋯ |
L(s) = 1 | + (−0.353 + 0.612i)2-s + (−0.418 − 0.908i)3-s + (−0.249 − 0.433i)4-s + 0.648·5-s + (0.704 + 0.0648i)6-s + 0.353·8-s + (−0.649 + 0.760i)9-s + (−0.229 + 0.396i)10-s + 0.603·11-s + (−0.288 + 0.408i)12-s + (0.679 − 1.17i)13-s + (−0.271 − 0.588i)15-s + (−0.125 + 0.216i)16-s + (0.242 − 0.420i)17-s + (−0.235 − 0.666i)18-s + (0.292 + 0.506i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.706 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.706 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.13775 - 0.471892i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.13775 - 0.471892i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 - 0.866i)T \) |
| 3 | \( 1 + (0.724 + 1.57i)T \) |
| 7 | \( 1 \) |
good | 5 | \( 1 - 1.44T + 5T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 13 | \( 1 + (-2.44 + 4.24i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-1 + 1.73i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.27 - 2.20i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + T + 23T^{2} \) |
| 29 | \( 1 + (-3.44 - 5.97i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-3 - 5.19i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (5.89 + 10.2i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-4.89 + 8.48i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (3.44 + 5.97i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-4.89 + 8.48i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-5.44 + 9.43i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (1 + 1.73i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-3.27 + 5.67i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-6.44 - 11.1i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 0.101T + 71T^{2} \) |
| 73 | \( 1 + (3.44 - 5.97i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-0.949 + 1.64i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-1 - 1.73i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (8.44 + 14.6i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-1.44 - 2.51i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−10.14175693847199419964701106993, −8.906249904374840855252433146203, −8.343849017400755660445553350682, −7.30547035772775785406513802899, −6.71087237881305448781180915965, −5.62341869987200928290279787606, −5.38346571977618304285884795067, −3.58864861722245537951804946080, −2.03965558342040984167070206358, −0.819332000966437229669241799166,
1.30875365332048548998681413404, 2.75820605462113403218118555636, 3.99054953000600804646729661251, 4.62975211292245217696290658794, 5.98652545964559830218429373690, 6.50700488671639483297729627346, 8.010583108700268794346219797067, 8.939340884868503815865812320803, 9.605867582969572173823233713285, 10.06099852652777235248427742959