L(s) = 1 | + (1 − i)2-s + (1.22 + 1.22i)3-s − 2i·4-s + 2.44·6-s + (1.87 − 1.87i)7-s + (−2 − 2i)8-s + 2.99i·9-s − 19.6·11-s + (2.44 − 2.44i)12-s + (0.747 + 0.747i)13-s − 3.74i·14-s − 4·16-s + (−20.6 + 20.6i)17-s + (2.99 + 2.99i)18-s + 13.7i·19-s + ⋯ |
L(s) = 1 | + (0.5 − 0.5i)2-s + (0.408 + 0.408i)3-s − 0.5i·4-s + 0.408·6-s + (0.267 − 0.267i)7-s + (−0.250 − 0.250i)8-s + 0.333i·9-s − 1.78·11-s + (0.204 − 0.204i)12-s + (0.0574 + 0.0574i)13-s − 0.267i·14-s − 0.250·16-s + (−1.21 + 1.21i)17-s + (0.166 + 0.166i)18-s + 0.721i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.130 - 0.991i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.130 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.367561164\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.367561164\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1 + i)T \) |
| 3 | \( 1 + (-1.22 - 1.22i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-1.87 + 1.87i)T \) |
good | 11 | \( 1 + 19.6T + 121T^{2} \) |
| 13 | \( 1 + (-0.747 - 0.747i)T + 169iT^{2} \) |
| 17 | \( 1 + (20.6 - 20.6i)T - 289iT^{2} \) |
| 19 | \( 1 - 13.7iT - 361T^{2} \) |
| 23 | \( 1 + (-15.2 - 15.2i)T + 529iT^{2} \) |
| 29 | \( 1 - 52.0iT - 841T^{2} \) |
| 31 | \( 1 - 6.68T + 961T^{2} \) |
| 37 | \( 1 + (-34.0 + 34.0i)T - 1.36e3iT^{2} \) |
| 41 | \( 1 + 2.98T + 1.68e3T^{2} \) |
| 43 | \( 1 + (-16.2 - 16.2i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 + (62.2 - 62.2i)T - 2.20e3iT^{2} \) |
| 53 | \( 1 + (21.3 + 21.3i)T + 2.80e3iT^{2} \) |
| 59 | \( 1 + 95.6iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 31.9T + 3.72e3T^{2} \) |
| 67 | \( 1 + (41.4 - 41.4i)T - 4.48e3iT^{2} \) |
| 71 | \( 1 - 108.T + 5.04e3T^{2} \) |
| 73 | \( 1 + (79.9 + 79.9i)T + 5.32e3iT^{2} \) |
| 79 | \( 1 - 87.2iT - 6.24e3T^{2} \) |
| 83 | \( 1 + (-17.2 - 17.2i)T + 6.88e3iT^{2} \) |
| 89 | \( 1 - 60.6iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (49.0 - 49.0i)T - 9.40e3iT^{2} \) |
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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
−10.12617547029012749150293792723, −9.243711788793009931850979682226, −8.288439441292010753426299148487, −7.62486758050364549426646844166, −6.42161983468541618609962454010, −5.35072970640850658451544918384, −4.67352930646998996895937991071, −3.65936777790417519352265573488, −2.72023888126736304881570115913, −1.63525431542195105391237405021,
0.30868803865896551725286423018, 2.40278516330522553583778945059, 2.85687468381133582332288005807, 4.47787662166048760851628961090, 5.07009555569761315622030198493, 6.13376851875567619883621229017, 7.03190826759645610786152959625, 7.77548004033462246939785448956, 8.448238809715563897827579946256, 9.271564372766634788717401562129