| L(s) = 1 | + 5.34i·3-s − 5.79·5-s − 5.87i·7-s − 19.5·9-s − 8.07i·11-s + 14.0·13-s − 30.9i·15-s + 29.9·17-s + 4.35i·19-s + 31.3·21-s + 8.74i·23-s + 8.55·25-s − 56.4i·27-s − 13.4·29-s − 7.65i·31-s + ⋯ |
| L(s) = 1 | + 1.78i·3-s − 1.15·5-s − 0.839i·7-s − 2.17·9-s − 0.734i·11-s + 1.07·13-s − 2.06i·15-s + 1.76·17-s + 0.229i·19-s + 1.49·21-s + 0.380i·23-s + 0.342·25-s − 2.08i·27-s − 0.463·29-s − 0.247i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.437773854\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.437773854\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 - 4.35iT \) |
| good | 3 | \( 1 - 5.34iT - 9T^{2} \) |
| 5 | \( 1 + 5.79T + 25T^{2} \) |
| 7 | \( 1 + 5.87iT - 49T^{2} \) |
| 11 | \( 1 + 8.07iT - 121T^{2} \) |
| 13 | \( 1 - 14.0T + 169T^{2} \) |
| 17 | \( 1 - 29.9T + 289T^{2} \) |
| 23 | \( 1 - 8.74iT - 529T^{2} \) |
| 29 | \( 1 + 13.4T + 841T^{2} \) |
| 31 | \( 1 + 7.65iT - 961T^{2} \) |
| 37 | \( 1 + 25.1T + 1.36e3T^{2} \) |
| 41 | \( 1 - 49.6T + 1.68e3T^{2} \) |
| 43 | \( 1 - 47.2iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 46.8iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 14.1T + 2.80e3T^{2} \) |
| 59 | \( 1 + 100. iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 31.1T + 3.72e3T^{2} \) |
| 67 | \( 1 + 34.5iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 66.0iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 27.5T + 5.32e3T^{2} \) |
| 79 | \( 1 - 40.9iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 87.8iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 28.6T + 7.92e3T^{2} \) |
| 97 | \( 1 - 9.35T + 9.40e3T^{2} \) |
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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
−9.825659540440807794446109112688, −9.058520782819183651501262995883, −8.110050405424465513776989197597, −7.65747347689107544814098961015, −6.15281543804523388413083188929, −5.34155179589614342486604293195, −4.31300359912566544651937795584, −3.60989866089067238145977042126, −3.31061451275249051567375751265, −0.811462581539802816348410841395,
0.65255691351828525687333279330, 1.75229345821980709796290311077, 2.87719954967988299493371366903, 3.89910049792394421162311796540, 5.42759647888224404971592111659, 6.05510420381512963533286714300, 7.19667624015113766633345125652, 7.49317683320774961186944653635, 8.415645369649368676690507854036, 8.843552244816824338359996688458
