L(s) = 1 | − 2.72i·2-s − 1.73·3-s − 3.44·4-s + 5.71·5-s + 4.72i·6-s + 8.69·7-s − 1.52i·8-s + 2.99·9-s − 15.5i·10-s − 6.01i·11-s + 5.95·12-s + 4.81i·13-s − 23.7i·14-s − 9.89·15-s − 17.9·16-s + 10.8·17-s + ⋯ |
L(s) = 1 | − 1.36i·2-s − 0.577·3-s − 0.860·4-s + 1.14·5-s + 0.787i·6-s + 1.24·7-s − 0.190i·8-s + 0.333·9-s − 1.55i·10-s − 0.546i·11-s + 0.496·12-s + 0.370i·13-s − 1.69i·14-s − 0.659·15-s − 1.12·16-s + 0.637·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.499 + 0.866i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.499 + 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.826459 - 1.43110i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.826459 - 1.43110i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + 1.73T \) |
| 59 | \( 1 + (-29.4 + 51.1i)T \) |
good | 2 | \( 1 + 2.72iT - 4T^{2} \) |
| 5 | \( 1 - 5.71T + 25T^{2} \) |
| 7 | \( 1 - 8.69T + 49T^{2} \) |
| 11 | \( 1 + 6.01iT - 121T^{2} \) |
| 13 | \( 1 - 4.81iT - 169T^{2} \) |
| 17 | \( 1 - 10.8T + 289T^{2} \) |
| 19 | \( 1 + 18.4T + 361T^{2} \) |
| 23 | \( 1 + 15.4iT - 529T^{2} \) |
| 29 | \( 1 - 15.5T + 841T^{2} \) |
| 31 | \( 1 - 6.38iT - 961T^{2} \) |
| 37 | \( 1 - 0.527iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 9.31T + 1.68e3T^{2} \) |
| 43 | \( 1 + 52.9iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 78.0iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 10.2T + 2.80e3T^{2} \) |
| 61 | \( 1 - 41.5iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 92.4iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 93.8T + 5.04e3T^{2} \) |
| 73 | \( 1 - 88.0iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 81.2T + 6.24e3T^{2} \) |
| 83 | \( 1 - 2.92iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 168. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 169. iT - 9.40e3T^{2} \) |
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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
−11.96632530932837102648165981028, −11.05583063793301130196522063025, −10.48413884581424227833716455880, −9.523179583802051758295639080027, −8.354171272585797884540583211299, −6.65107589553893862308959890785, −5.44277750215200698952039499322, −4.23285799665214673766072422957, −2.39387978226828740447239763181, −1.25948377869023240248073928998,
1.89412007071514535499698581334, 4.69279653307237539847376860665, 5.50228858669751730570906226228, 6.34911328458730837372080835135, 7.49631242874829311250775902990, 8.408535147425674734088687678472, 9.675514736956391462975609473827, 10.73553997191787105615948129158, 11.78847878157488403737483749405, 13.06555125164158512117664437800