| L(s) = 1 | − 3.73i·2-s + 2.01·4-s − 11.1i·5-s + 39.4·7-s − 67.3i·8-s − 41.8·10-s + 194. i·11-s − 96.0·13-s − 147. i·14-s − 219.·16-s − 404. i·17-s − 559.·19-s − 22.5i·20-s + 728.·22-s − 780. i·23-s + ⋯ |
| L(s) = 1 | − 0.934i·2-s + 0.126·4-s − 0.447i·5-s + 0.804·7-s − 1.05i·8-s − 0.418·10-s + 1.61i·11-s − 0.568·13-s − 0.751i·14-s − 0.857·16-s − 1.39i·17-s − 1.54·19-s − 0.0564i·20-s + 1.50·22-s − 1.47i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.698309620\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.698309620\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + 11.1iT \) |
| good | 2 | \( 1 + 3.73iT - 16T^{2} \) |
| 7 | \( 1 - 39.4T + 2.40e3T^{2} \) |
| 11 | \( 1 - 194. iT - 1.46e4T^{2} \) |
| 13 | \( 1 + 96.0T + 2.85e4T^{2} \) |
| 17 | \( 1 + 404. iT - 8.35e4T^{2} \) |
| 19 | \( 1 + 559.T + 1.30e5T^{2} \) |
| 23 | \( 1 + 780. iT - 2.79e5T^{2} \) |
| 29 | \( 1 + 478. iT - 7.07e5T^{2} \) |
| 31 | \( 1 - 622.T + 9.23e5T^{2} \) |
| 37 | \( 1 - 1.40e3T + 1.87e6T^{2} \) |
| 41 | \( 1 + 100. iT - 2.82e6T^{2} \) |
| 43 | \( 1 + 598.T + 3.41e6T^{2} \) |
| 47 | \( 1 + 2.13e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 + 2.45e3iT - 7.89e6T^{2} \) |
| 59 | \( 1 + 4.17e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 - 691.T + 1.38e7T^{2} \) |
| 67 | \( 1 + 5.43e3T + 2.01e7T^{2} \) |
| 71 | \( 1 - 199. iT - 2.54e7T^{2} \) |
| 73 | \( 1 + 5.15e3T + 2.83e7T^{2} \) |
| 79 | \( 1 + 5.53e3T + 3.89e7T^{2} \) |
| 83 | \( 1 - 952. iT - 4.74e7T^{2} \) |
| 89 | \( 1 - 8.55e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 + 5.11e3T + 8.85e7T^{2} \) |
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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
−10.18875956446321669173989314542, −9.623332995686760722204690604918, −8.445149506394672257208718288488, −7.36545382395647712102945570217, −6.52034528089159174012367371011, −4.80616836652594174256298058966, −4.34069112152252836757972605417, −2.55107968720205861653720107506, −1.88257221340627612596175234893, −0.43268447335937283607040419827,
1.62529673935207366584335480125, 2.97458778220005020176562364585, 4.41474326189660329052501321031, 5.76657684650770927718034849783, 6.21639200136508973277260796815, 7.42039678740021540868843795893, 8.181141035034467642743931190307, 8.821308643615657394044750144869, 10.40203100373139118218526739840, 11.08174430164396092560248769391
