L(s) = 1 | − 0.816i·2-s + 3.33·4-s + 1.04i·5-s + 6.83·7-s − 5.98i·8-s + 0.855·10-s − 7.70·13-s − 5.57i·14-s + 8.44·16-s − 9.39i·17-s + 9.42·19-s + 3.48i·20-s − 4.82i·23-s + 23.9·25-s + 6.29i·26-s + ⋯ |
L(s) = 1 | − 0.408i·2-s + 0.833·4-s + 0.209i·5-s + 0.975·7-s − 0.748i·8-s + 0.0855·10-s − 0.592·13-s − 0.398i·14-s + 0.527·16-s − 0.552i·17-s + 0.495·19-s + 0.174i·20-s − 0.209i·23-s + 0.956·25-s + 0.242i·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(2.781760202\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.781760202\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + 0.816iT - 4T^{2} \) |
| 5 | \( 1 - 1.04iT - 25T^{2} \) |
| 7 | \( 1 - 6.83T + 49T^{2} \) |
| 13 | \( 1 + 7.70T + 169T^{2} \) |
| 17 | \( 1 + 9.39iT - 289T^{2} \) |
| 19 | \( 1 - 9.42T + 361T^{2} \) |
| 23 | \( 1 + 4.82iT - 529T^{2} \) |
| 29 | \( 1 + 46.7iT - 841T^{2} \) |
| 31 | \( 1 - 18.9T + 961T^{2} \) |
| 37 | \( 1 + 47.1T + 1.36e3T^{2} \) |
| 41 | \( 1 + 7.60iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 50.6T + 1.84e3T^{2} \) |
| 47 | \( 1 - 66.1iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 38.4iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 117. iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 81.4T + 3.72e3T^{2} \) |
| 67 | \( 1 - 70.1T + 4.48e3T^{2} \) |
| 71 | \( 1 - 43.2iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 45.9T + 5.32e3T^{2} \) |
| 79 | \( 1 + 85.7T + 6.24e3T^{2} \) |
| 83 | \( 1 + 159. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 154. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 153.T + 9.40e3T^{2} \) |
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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
−9.798413601117052392447132939721, −8.725078140169516825128632860788, −7.72282969033899613302264245450, −7.18710592593775945706310871517, −6.22203617046905608445926827964, −5.19835982354244041745318852133, −4.23416592832535609252896461775, −2.95482612528632181258338548807, −2.16733761285764479809092957271, −0.931181589979163827958812987567,
1.28137620415238939486028493012, 2.30464763078227721763787827287, 3.51445522522545704463781561112, 4.95213532631769754582753429600, 5.38844020765326056055185938353, 6.62872370870909019705756076595, 7.23246156705445401885136927335, 8.123570170547660873037899131698, 8.689382137456482671667620172775, 9.850937394611234421157020938680