| L(s) = 1 | − 1.41i·2-s + 1.70·3-s − 2.00·4-s − 2.23·5-s − 2.41i·6-s − 2.53i·7-s + 2.82i·8-s − 6.08·9-s + 3.16i·10-s − 3.41·12-s − 13.4i·13-s − 3.58·14-s − 3.82·15-s + 4.00·16-s + 30.5i·17-s + 8.60i·18-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s + 0.569·3-s − 0.500·4-s − 0.447·5-s − 0.402i·6-s − 0.361i·7-s + 0.353i·8-s − 0.675·9-s + 0.316i·10-s − 0.284·12-s − 1.03i·13-s − 0.255·14-s − 0.254·15-s + 0.250·16-s + 1.79i·17-s + 0.477i·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.219 - 0.975i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.219 - 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.5767622790\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5767622790\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 1.41iT \) |
| 5 | \( 1 + 2.23T \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 - 1.70T + 9T^{2} \) |
| 7 | \( 1 + 2.53iT - 49T^{2} \) |
| 13 | \( 1 + 13.4iT - 169T^{2} \) |
| 17 | \( 1 - 30.5iT - 289T^{2} \) |
| 19 | \( 1 + 32.2iT - 361T^{2} \) |
| 23 | \( 1 + 0.676T + 529T^{2} \) |
| 29 | \( 1 - 34.1iT - 841T^{2} \) |
| 31 | \( 1 - 10.8T + 961T^{2} \) |
| 37 | \( 1 + 29.5T + 1.36e3T^{2} \) |
| 41 | \( 1 - 68.3iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 28.1iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 19.9T + 2.20e3T^{2} \) |
| 53 | \( 1 + 15.2T + 2.80e3T^{2} \) |
| 59 | \( 1 + 105.T + 3.48e3T^{2} \) |
| 61 | \( 1 - 106. iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 44.0T + 4.48e3T^{2} \) |
| 71 | \( 1 + 12.5T + 5.04e3T^{2} \) |
| 73 | \( 1 - 37.8iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 48.1iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 92.2iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 107.T + 7.92e3T^{2} \) |
| 97 | \( 1 - 33.0T + 9.40e3T^{2} \) |
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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
−9.762038784251216457795480148402, −8.783156427589137830558976519995, −8.336486921959172082522733744001, −7.52750821302863369307608318886, −6.38563939454902838246141657131, −5.31182438435340416973584718166, −4.31585592833600178303557875031, −3.33194047050073034650635059027, −2.70148119250009041405923468727, −1.25465020888553274822424083429,
0.16316932313486827862080786080, 2.05198171595705743316523158059, 3.23999533835553520050315147501, 4.15889352379002548483831227586, 5.21387233200386526155177983403, 6.03951869560669356822337243934, 7.02915933090930695202251508312, 7.77917453210894252643051580729, 8.483315453661602074115601056552, 9.201019170216745880666739189759
