| L(s) = 1 | + (4.25 + 2.63i)5-s + 2.72i·7-s + 3.20i·11-s − 7.99i·13-s + 1.82·17-s − 15.4·19-s + 32.3·23-s + (11.1 + 22.3i)25-s − 29.5i·29-s + 53.0·31-s + (−7.16 + 11.5i)35-s + 30.0i·37-s + 76.8i·41-s − 8.98i·43-s − 11.8·47-s + ⋯ |
| L(s) = 1 | + (0.850 + 0.526i)5-s + 0.388i·7-s + 0.291i·11-s − 0.614i·13-s + 0.107·17-s − 0.812·19-s + 1.40·23-s + (0.445 + 0.895i)25-s − 1.01i·29-s + 1.70·31-s + (−0.204 + 0.330i)35-s + 0.812i·37-s + 1.87i·41-s − 0.208i·43-s − 0.252·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.526 - 0.850i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.526 - 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.361613839\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.361613839\) |
| \(L(2)\) |
|
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 | \( 1 + (-4.25 - 2.63i)T \) |
| good | 7 | \( 1 - 2.72iT - 49T^{2} \) |
| 11 | \( 1 - 3.20iT - 121T^{2} \) |
| 13 | \( 1 + 7.99iT - 169T^{2} \) |
| 17 | \( 1 - 1.82T + 289T^{2} \) |
| 19 | \( 1 + 15.4T + 361T^{2} \) |
| 23 | \( 1 - 32.3T + 529T^{2} \) |
| 29 | \( 1 + 29.5iT - 841T^{2} \) |
| 31 | \( 1 - 53.0T + 961T^{2} \) |
| 37 | \( 1 - 30.0iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 76.8iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 8.98iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 11.8T + 2.20e3T^{2} \) |
| 53 | \( 1 - 28.9T + 2.80e3T^{2} \) |
| 59 | \( 1 - 19.2iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 61.9T + 3.72e3T^{2} \) |
| 67 | \( 1 - 5.18iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 96.0iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 127. iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 48.4T + 6.24e3T^{2} \) |
| 83 | \( 1 + 96.0T + 6.88e3T^{2} \) |
| 89 | \( 1 - 21.4iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 166. iT - 9.40e3T^{2} \) |
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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
−9.449581933030519455277381282889, −8.569676194401521531789408928195, −7.79115544556653910675069052629, −6.72647938903718543831277948585, −6.20666307323174917560393914487, −5.29483590119298315504332121218, −4.44317215153090181199051381991, −3.04975565835602185459251772252, −2.43432338754093577295105894149, −1.12919247683588379718826963784,
0.71826840147510488873525673207, 1.81837479827262288141925614916, 2.91860643863063866685526105961, 4.15781737309910395971170345124, 4.94417955936082564214837776343, 5.81392959024182871709410301071, 6.65325269858368292443354702057, 7.36332042501005693060706547229, 8.652773812237421250772007154998, 8.881952584441739772902110662012
