| L(s) = 1 | + (0.161 + 1.53i)2-s + (−0.382 + 0.0811i)4-s + (0.0618 − 0.588i)5-s + (0.483 + 0.536i)7-s + (0.768 + 2.36i)8-s + 0.915·10-s + (2.56 + 2.10i)11-s + (−1.43 − 0.640i)13-s + (−0.746 + 0.829i)14-s + (−4.22 + 1.88i)16-s + (3.71 + 2.69i)17-s + (0.775 + 2.38i)19-s + (0.0241 + 0.229i)20-s + (−2.82 + 4.28i)22-s + (−2.22 − 3.85i)23-s + ⋯ |
| L(s) = 1 | + (0.114 + 1.08i)2-s + (−0.191 + 0.0405i)4-s + (0.0276 − 0.263i)5-s + (0.182 + 0.202i)7-s + (0.271 + 0.836i)8-s + 0.289·10-s + (0.773 + 0.634i)11-s + (−0.399 − 0.177i)13-s + (−0.199 + 0.221i)14-s + (−1.05 + 0.470i)16-s + (0.901 + 0.654i)17-s + (0.177 + 0.547i)19-s + (0.00540 + 0.0514i)20-s + (−0.601 + 0.913i)22-s + (−0.464 − 0.803i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 297 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0768 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 297 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0768 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.05436 + 1.13876i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.05436 + 1.13876i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 + (-2.56 - 2.10i)T \) |
| good | 2 | \( 1 + (-0.161 - 1.53i)T + (-1.95 + 0.415i)T^{2} \) |
| 5 | \( 1 + (-0.0618 + 0.588i)T + (-4.89 - 1.03i)T^{2} \) |
| 7 | \( 1 + (-0.483 - 0.536i)T + (-0.731 + 6.96i)T^{2} \) |
| 13 | \( 1 + (1.43 + 0.640i)T + (8.69 + 9.66i)T^{2} \) |
| 17 | \( 1 + (-3.71 - 2.69i)T + (5.25 + 16.1i)T^{2} \) |
| 19 | \( 1 + (-0.775 - 2.38i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + (2.22 + 3.85i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (4.66 + 5.18i)T + (-3.03 + 28.8i)T^{2} \) |
| 31 | \( 1 + (8.33 + 3.71i)T + (20.7 + 23.0i)T^{2} \) |
| 37 | \( 1 + (-0.893 + 2.75i)T + (-29.9 - 21.7i)T^{2} \) |
| 41 | \( 1 + (-0.772 + 0.857i)T + (-4.28 - 40.7i)T^{2} \) |
| 43 | \( 1 + (-2.10 + 3.64i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (0.222 + 0.0473i)T + (42.9 + 19.1i)T^{2} \) |
| 53 | \( 1 + (4.61 - 3.35i)T + (16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (-6.97 + 1.48i)T + (53.8 - 23.9i)T^{2} \) |
| 61 | \( 1 + (-4.16 + 1.85i)T + (40.8 - 45.3i)T^{2} \) |
| 67 | \( 1 + (-4.04 - 7.00i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (9.95 + 7.23i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (-4.78 + 14.7i)T + (-59.0 - 42.9i)T^{2} \) |
| 79 | \( 1 + (-1.52 - 14.5i)T + (-77.2 + 16.4i)T^{2} \) |
| 83 | \( 1 + (-6.35 + 2.82i)T + (55.5 - 61.6i)T^{2} \) |
| 89 | \( 1 + 12.4T + 89T^{2} \) |
| 97 | \( 1 + (1.48 + 14.1i)T + (-94.8 + 20.1i)T^{2} \) |
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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
−12.12570871960175688940098567335, −11.12344397735371212821171723073, −9.993811198208170341402713415836, −8.937789953667192319098578564162, −7.936499774741107766434260083048, −7.17782456215027753602991157245, −6.06970310788776742673109305510, −5.26403063495356088363402200951, −3.99485803099802237803374090488, −1.99812667945740083865842198627,
1.33726877035637819449937128583, 2.91297947744653486274736301593, 3.84341003433951708751758765932, 5.26556758169028005686574277799, 6.72660232591305429426967139206, 7.53523528493893281878380018017, 9.045470917691270380228457417826, 9.794196945823787884104841826524, 10.87694317187224996045117364228, 11.40374971133478305286919864809
