| L(s) = 1 | − 2.78i·2-s − 5.54·3-s − 3.74·4-s + 15.4i·6-s + 4.48i·7-s − 0.713i·8-s + 21.7·9-s + (2.46 + 10.7i)11-s + 20.7·12-s + 15.7i·13-s + 12.4·14-s − 16.9·16-s − 25.4i·17-s − 60.5i·18-s − 8.44i·19-s + ⋯ |
| L(s) = 1 | − 1.39i·2-s − 1.84·3-s − 0.935·4-s + 2.57i·6-s + 0.640i·7-s − 0.0892i·8-s + 2.41·9-s + (0.223 + 0.974i)11-s + 1.73·12-s + 1.20i·13-s + 0.890·14-s − 1.06·16-s − 1.49i·17-s − 3.36i·18-s − 0.444i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.223 + 0.974i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.223 + 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.646305 - 0.514640i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.646305 - 0.514640i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 11 | \( 1 + (-2.46 - 10.7i)T \) |
| good | 2 | \( 1 + 2.78iT - 4T^{2} \) |
| 3 | \( 1 + 5.54T + 9T^{2} \) |
| 7 | \( 1 - 4.48iT - 49T^{2} \) |
| 13 | \( 1 - 15.7iT - 169T^{2} \) |
| 17 | \( 1 + 25.4iT - 289T^{2} \) |
| 19 | \( 1 + 8.44iT - 361T^{2} \) |
| 23 | \( 1 - 19.8T + 529T^{2} \) |
| 29 | \( 1 + 21.4iT - 841T^{2} \) |
| 31 | \( 1 - 24.9T + 961T^{2} \) |
| 37 | \( 1 - 30.4T + 1.36e3T^{2} \) |
| 41 | \( 1 - 10.5iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 68.3iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 25.0T + 2.20e3T^{2} \) |
| 53 | \( 1 - 93.9T + 2.80e3T^{2} \) |
| 59 | \( 1 + 27.3T + 3.48e3T^{2} \) |
| 61 | \( 1 + 81.1iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 49.5T + 4.48e3T^{2} \) |
| 71 | \( 1 - 44.0T + 5.04e3T^{2} \) |
| 73 | \( 1 + 29.1iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 51.3iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 47.2iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 147.T + 7.92e3T^{2} \) |
| 97 | \( 1 + 87.8T + 9.40e3T^{2} \) |
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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
−11.58891178759872076239545446556, −10.93980369665896898294955174189, −9.770657716309100998853376322112, −9.351276019942880202848226349649, −7.15541020626768120167912734182, −6.42392740352274474373020704204, −4.99020151498800782043755367634, −4.33181489078193321105528487617, −2.35847132990586069169802164956, −0.899255103714946212744567680969,
0.793474208842541099028462445184, 4.03133359282241071070954843723, 5.34234099609566344836815531653, 5.86487373235578785588161515877, 6.69689352247044700435797593014, 7.58802370442786134211939151874, 8.659108681650286383479172733802, 10.39865165461062282518470388945, 10.74110883239418601817066343188, 11.81754677320406281607061429628
