L(s) = 1 | + 1.41·2-s + (−1.41 − 2.64i)3-s + 2.00·4-s + (−2.00 − 3.74i)6-s − 2.64i·7-s + 2.82·8-s + (−5 + 7.48i)9-s + 14.5i·11-s + (−2.82 − 5.29i)12-s + 0.583i·13-s − 3.74i·14-s + 4.00·16-s − 21.5·17-s + (−7.07 + 10.5i)18-s + 16·19-s + ⋯ |
L(s) = 1 | + 0.707·2-s + (−0.471 − 0.881i)3-s + 0.500·4-s + (−0.333 − 0.623i)6-s − 0.377i·7-s + 0.353·8-s + (−0.555 + 0.831i)9-s + 1.32i·11-s + (−0.235 − 0.440i)12-s + 0.0448i·13-s − 0.267i·14-s + 0.250·16-s − 1.26·17-s + (−0.392 + 0.587i)18-s + 0.842·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0272 - 0.999i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0272 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.094567206\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.094567206\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - 1.41T \) |
| 3 | \( 1 + (1.41 + 2.64i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + 2.64iT \) |
good | 11 | \( 1 - 14.5iT - 121T^{2} \) |
| 13 | \( 1 - 0.583iT - 169T^{2} \) |
| 17 | \( 1 + 21.5T + 289T^{2} \) |
| 19 | \( 1 - 16T + 361T^{2} \) |
| 23 | \( 1 + 38.8T + 529T^{2} \) |
| 29 | \( 1 - 35.7iT - 841T^{2} \) |
| 31 | \( 1 + 58.4T + 961T^{2} \) |
| 37 | \( 1 - 20iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 8.75iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 11.7iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 8.48T + 2.20e3T^{2} \) |
| 53 | \( 1 - 50.9T + 2.80e3T^{2} \) |
| 59 | \( 1 - 58.2iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 38.9T + 3.72e3T^{2} \) |
| 67 | \( 1 - 70.5iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 17.0iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 72.3iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 20.9T + 6.24e3T^{2} \) |
| 83 | \( 1 - 145.T + 6.88e3T^{2} \) |
| 89 | \( 1 - 53.6iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 111. iT - 9.40e3T^{2} \) |
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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
−10.16474571510322760569371585843, −9.086748486839385250364536703927, −7.906199179781759832767656629005, −7.18640216193214247065083372576, −6.66227924266584638743518526637, −5.62570478643244042451123098901, −4.81276826271868333764230218880, −3.84743393079751880094468931402, −2.39007065924449031750385939930, −1.53687802565022089856726828604,
0.26282220133297452365488722817, 2.24843877704961615214375485095, 3.49879008959662170751897412718, 4.11919327145417163132428644427, 5.30022055774376282027502878222, 5.83127239954032311393109053651, 6.59565673075523889183082741798, 7.88055693315394199823466902410, 8.814653875831354721789978996381, 9.563016390471159302585051534395