| L(s) = 1 | − 2.94·5-s − 5.84·7-s + 18.6·11-s − 1.55i·13-s − 12.7·17-s + (17.7 + 6.84i)19-s − 15.8·23-s − 16.2·25-s − 46.0i·29-s + 37.5i·31-s + 17.2·35-s + 17.6i·37-s − 33.3i·41-s + 50.6·43-s − 15.1·47-s + ⋯ |
| L(s) = 1 | − 0.589·5-s − 0.835·7-s + 1.69·11-s − 0.119i·13-s − 0.749·17-s + (0.932 + 0.360i)19-s − 0.687·23-s − 0.651·25-s − 1.58i·29-s + 1.21i·31-s + 0.492·35-s + 0.477i·37-s − 0.814i·41-s + 1.17·43-s − 0.322·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.932 + 0.360i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.932 + 0.360i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.511917581\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.511917581\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 + (-17.7 - 6.84i)T \) |
| good | 5 | \( 1 + 2.94T + 25T^{2} \) |
| 7 | \( 1 + 5.84T + 49T^{2} \) |
| 11 | \( 1 - 18.6T + 121T^{2} \) |
| 13 | \( 1 + 1.55iT - 169T^{2} \) |
| 17 | \( 1 + 12.7T + 289T^{2} \) |
| 23 | \( 1 + 15.8T + 529T^{2} \) |
| 29 | \( 1 + 46.0iT - 841T^{2} \) |
| 31 | \( 1 - 37.5iT - 961T^{2} \) |
| 37 | \( 1 - 17.6iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 33.3iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 50.6T + 1.84e3T^{2} \) |
| 47 | \( 1 + 15.1T + 2.20e3T^{2} \) |
| 53 | \( 1 - 75.3iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 53.0iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 11.3T + 3.72e3T^{2} \) |
| 67 | \( 1 + 12.1iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 20.8iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 34.2T + 5.32e3T^{2} \) |
| 79 | \( 1 - 128. iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 66.9T + 6.88e3T^{2} \) |
| 89 | \( 1 + 25.1iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 117. iT - 9.40e3T^{2} \) |
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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
−8.634092266429866962828535467811, −7.83837058058327935775253219126, −7.01614344519840590604675526176, −6.36876543883223614497933738547, −5.70108226096814398708985518107, −4.38546583578347628921605488775, −3.86961021822894680352826669883, −3.05444668596150923816355466758, −1.76916968118187837877023604976, −0.54878309901546151222170478053,
0.69220142416629661380447346930, 1.91704320301905132817471131607, 3.20891902588254496732685627797, 3.85722539109969554890174457425, 4.56910656798448250676906851336, 5.77554770319695769633291651247, 6.48792843893478784320126779202, 7.09511139889289003322156503786, 7.86967091165293771391614696114, 8.892790793015743459961289913964
