| L(s) = 1 | + (1.73 − i)2-s + (1.99 − 3.46i)4-s − 9.38i·7-s − 7.99i·8-s + 16.2i·11-s + 16.2·13-s + (−9.38 − 16.2i)14-s + (−8 − 13.8i)16-s + 10.3·17-s − 20.7i·19-s + (16.2 + 28.1i)22-s − 28i·23-s + (28.1 − 16.2i)26-s + (−32.4 − 18.7i)28-s − 9.38·29-s + ⋯ |
| L(s) = 1 | + (0.866 − 0.5i)2-s + (0.499 − 0.866i)4-s − 1.34i·7-s − 0.999i·8-s + 1.47i·11-s + 1.24·13-s + (−0.670 − 1.16i)14-s + (−0.5 − 0.866i)16-s + 0.611·17-s − 1.09i·19-s + (0.738 + 1.27i)22-s − 1.21i·23-s + (1.08 − 0.624i)26-s + (−1.16 − 0.670i)28-s − 0.323·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.5 + 0.866i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.5 + 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(3.163522485\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.163522485\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.73 + i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 9.38iT - 49T^{2} \) |
| 11 | \( 1 - 16.2iT - 121T^{2} \) |
| 13 | \( 1 - 16.2T + 169T^{2} \) |
| 17 | \( 1 - 10.3T + 289T^{2} \) |
| 19 | \( 1 + 20.7iT - 361T^{2} \) |
| 23 | \( 1 + 28iT - 529T^{2} \) |
| 29 | \( 1 + 9.38T + 841T^{2} \) |
| 31 | \( 1 + 34.6iT - 961T^{2} \) |
| 37 | \( 1 + 48.7T + 1.36e3T^{2} \) |
| 41 | \( 1 + 18.7T + 1.68e3T^{2} \) |
| 43 | \( 1 - 37.5iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 4iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 31.1T + 2.80e3T^{2} \) |
| 59 | \( 1 - 16.2iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 58T + 3.72e3T^{2} \) |
| 67 | \( 1 - 18.7iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 97.4iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 5.32e3T^{2} \) |
| 79 | \( 1 - 6.92iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 32iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 75.0T + 7.92e3T^{2} \) |
| 97 | \( 1 - 162.T + 9.40e3T^{2} \) |
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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
−10.00700626945023284623367503820, −9.006121250795394418616807862956, −7.64133361469270196985708057986, −6.94809052376465214855987796430, −6.16491746334045464914686904031, −4.86350303710798287807375936376, −4.25409689649732281668595662552, −3.34940781639855689042821517675, −1.98039305668382624059394230539, −0.78322721863248376204471309282,
1.70072472925701155522409016995, 3.21406249644019628782353713513, 3.64434795606554861693129735083, 5.32998778987126570310926674177, 5.70489791343400368921189291498, 6.41759998447519334505766998920, 7.65465225089796972244571440640, 8.593355045274003085001557391390, 8.852604777798624431470048041930, 10.36888586938540723270088096783
