| L(s) = 1 | + (2.98 + 0.323i)3-s + 4.72·7-s + (8.79 + 1.92i)9-s + 4.76i·11-s + 1.06·13-s − 26.7i·17-s + 8.12·19-s + (14.0 + 1.52i)21-s − 40.0i·23-s + (25.5 + 8.59i)27-s + 20.8i·29-s + 33.7·31-s + (−1.53 + 14.2i)33-s + 60.4·37-s + (3.18 + 0.344i)39-s + ⋯ |
| L(s) = 1 | + (0.994 + 0.107i)3-s + 0.675·7-s + (0.976 + 0.214i)9-s + 0.433i·11-s + 0.0820·13-s − 1.57i·17-s + 0.427·19-s + (0.671 + 0.0727i)21-s − 1.74i·23-s + (0.948 + 0.318i)27-s + 0.719i·29-s + 1.08·31-s + (−0.0466 + 0.430i)33-s + 1.63·37-s + (0.0815 + 0.00884i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 + 0.107i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.994 + 0.107i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(3.300938834\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.300938834\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-2.98 - 0.323i)T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 4.72T + 49T^{2} \) |
| 11 | \( 1 - 4.76iT - 121T^{2} \) |
| 13 | \( 1 - 1.06T + 169T^{2} \) |
| 17 | \( 1 + 26.7iT - 289T^{2} \) |
| 19 | \( 1 - 8.12T + 361T^{2} \) |
| 23 | \( 1 + 40.0iT - 529T^{2} \) |
| 29 | \( 1 - 20.8iT - 841T^{2} \) |
| 31 | \( 1 - 33.7T + 961T^{2} \) |
| 37 | \( 1 - 60.4T + 1.36e3T^{2} \) |
| 41 | \( 1 - 59.2iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 56.4T + 1.84e3T^{2} \) |
| 47 | \( 1 + 9.68iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 93.1iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 17.4iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 57.7T + 3.72e3T^{2} \) |
| 67 | \( 1 - 101.T + 4.48e3T^{2} \) |
| 71 | \( 1 + 90.1iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 40.0T + 5.32e3T^{2} \) |
| 79 | \( 1 + 65.3T + 6.24e3T^{2} \) |
| 83 | \( 1 + 117. iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 119. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 15.2T + 9.40e3T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.550421813344086200871604344804, −8.653860073210186030414426703610, −8.006262700442720525771298393265, −7.24646322844808825129247628440, −6.39694215672444438458229935546, −4.85496957991313082843950035558, −4.53806882473424050398305182043, −3.12353016205193154549720237508, −2.36913915301760801065942612228, −1.05215836953936899795533512131,
1.21221949460811362331580289792, 2.18214405105012144347760406153, 3.42506157615034625801458846735, 4.12458808359885934867262797495, 5.30064125446870612044541557213, 6.29078946069895515294336806764, 7.31741623405538878961845579973, 8.188602278572910382574111885300, 8.448128222909987350237768464267, 9.610475893923936159799319720620
