| L(s) = 1 | + (2.12 + 4.52i)5-s + 4.02i·7-s + 11.5i·11-s + 13.5i·13-s − 4.76·17-s − 8.28·19-s − 33.4·23-s + (−15.9 + 19.2i)25-s − 21.3i·29-s − 51.2·31-s + (−18.2 + 8.54i)35-s − 1.30i·37-s + 0.277i·41-s − 5.16i·43-s + 49.9·47-s + ⋯ |
| L(s) = 1 | + (0.424 + 0.905i)5-s + 0.574i·7-s + 1.05i·11-s + 1.04i·13-s − 0.280·17-s − 0.435·19-s − 1.45·23-s + (−0.638 + 0.769i)25-s − 0.735i·29-s − 1.65·31-s + (−0.520 + 0.244i)35-s − 0.0351i·37-s + 0.00676i·41-s − 0.120i·43-s + 1.06·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.905 + 0.424i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.905 + 0.424i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.8121572028\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8121572028\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-2.12 - 4.52i)T \) |
| good | 7 | \( 1 - 4.02iT - 49T^{2} \) |
| 11 | \( 1 - 11.5iT - 121T^{2} \) |
| 13 | \( 1 - 13.5iT - 169T^{2} \) |
| 17 | \( 1 + 4.76T + 289T^{2} \) |
| 19 | \( 1 + 8.28T + 361T^{2} \) |
| 23 | \( 1 + 33.4T + 529T^{2} \) |
| 29 | \( 1 + 21.3iT - 841T^{2} \) |
| 31 | \( 1 + 51.2T + 961T^{2} \) |
| 37 | \( 1 + 1.30iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 0.277iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 5.16iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 49.9T + 2.20e3T^{2} \) |
| 53 | \( 1 + 14.5T + 2.80e3T^{2} \) |
| 59 | \( 1 + 56.4iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 108.T + 3.72e3T^{2} \) |
| 67 | \( 1 + 61.7iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 63.6iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 3.03iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 24.1T + 6.24e3T^{2} \) |
| 83 | \( 1 + 54.1T + 6.88e3T^{2} \) |
| 89 | \( 1 - 19.2iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 179. iT - 9.40e3T^{2} \) |
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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
−9.468942139186550315596957911535, −8.664066791772652052125381359855, −7.62793865620666754135447029277, −6.97615306266722119683161832796, −6.24933430482178629646097186096, −5.53153331500417462745243457582, −4.41017376121900751276880097266, −3.64451358214034109281132432799, −2.20766979950709360168945402489, −2.00528621537479012284813739118,
0.19687226707951845148685151685, 1.17360126126974376797046651840, 2.36959833930904987349083841007, 3.60668067750953431154251723023, 4.30898883920631730744498620731, 5.54259075627942610497419023492, 5.75037831904661811218482381715, 6.91619137693623628867058378812, 7.83124468597332589317321755004, 8.509219732143335628058264221750
