| L(s) = 1 | + 1.66·2-s − 2.79i·3-s − 1.21·4-s − 4.66i·6-s − 6.72·7-s − 8.70·8-s + 1.18·9-s + (−6.62 − 8.78i)11-s + 3.40i·12-s + 2.91·13-s − 11.2·14-s − 9.63·16-s − 30.9·17-s + 1.98·18-s − 4.49i·19-s + ⋯ |
| L(s) = 1 | + 0.833·2-s − 0.931i·3-s − 0.304·4-s − 0.776i·6-s − 0.960·7-s − 1.08·8-s + 0.131·9-s + (−0.601 − 0.798i)11-s + 0.283i·12-s + 0.224·13-s − 0.800·14-s − 0.602·16-s − 1.82·17-s + 0.110·18-s − 0.236i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.983 + 0.181i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.983 + 0.181i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.0818965 - 0.896152i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0818965 - 0.896152i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 11 | \( 1 + (6.62 + 8.78i)T \) |
| good | 2 | \( 1 - 1.66T + 4T^{2} \) |
| 3 | \( 1 + 2.79iT - 9T^{2} \) |
| 7 | \( 1 + 6.72T + 49T^{2} \) |
| 13 | \( 1 - 2.91T + 169T^{2} \) |
| 17 | \( 1 + 30.9T + 289T^{2} \) |
| 19 | \( 1 + 4.49iT - 361T^{2} \) |
| 23 | \( 1 + 13.0iT - 529T^{2} \) |
| 29 | \( 1 - 44.8iT - 841T^{2} \) |
| 31 | \( 1 - 26.1T + 961T^{2} \) |
| 37 | \( 1 + 28.8iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 24.6iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 2.28T + 1.84e3T^{2} \) |
| 47 | \( 1 + 2.33iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 93.1iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 108.T + 3.48e3T^{2} \) |
| 61 | \( 1 + 54.7iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 9.01iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 1.24T + 5.04e3T^{2} \) |
| 73 | \( 1 + 24.1T + 5.32e3T^{2} \) |
| 79 | \( 1 + 108. iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 8.27T + 6.88e3T^{2} \) |
| 89 | \( 1 + 84.8T + 7.92e3T^{2} \) |
| 97 | \( 1 - 160. iT - 9.40e3T^{2} \) |
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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
−11.52510553951838244286588236098, −10.39992165242319005535676500586, −9.131769334207694981124526241251, −8.356945204115947523839703596257, −6.86191579351028613533686218728, −6.31276940410074096412042064919, −5.06222159509695116332396391741, −3.77956873197969127295121412126, −2.52974850334629476591426333383, −0.33262634963906608868709399160,
2.75465699422453379407047507988, 4.07425000618941614971917336043, 4.61998705727396300448059226787, 5.86385426467432522285736743037, 6.91667728748341508033817771967, 8.485430788713509622830128752777, 9.583989065195657670794823358049, 9.973285636153647444860952316607, 11.20725352173549654495959964227, 12.33214257643929452398503731810
