| L(s) = 1 | − 6.42·5-s + 13.8i·7-s + 13.0i·11-s − 7.16·13-s + 31.5·17-s + 16.4i·19-s + 16.9i·23-s + 16.2·25-s + 42.4·29-s + 29.6i·31-s − 88.6i·35-s − 39.3·37-s + 39.8·41-s + 16.3i·43-s + 57.8i·47-s + ⋯ |
| L(s) = 1 | − 1.28·5-s + 1.97i·7-s + 1.18i·11-s − 0.551·13-s + 1.85·17-s + 0.867i·19-s + 0.738i·23-s + 0.649·25-s + 1.46·29-s + 0.957i·31-s − 2.53i·35-s − 1.06·37-s + 0.972·41-s + 0.379i·43-s + 1.23i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.133859906\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.133859906\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 6.42T + 25T^{2} \) |
| 7 | \( 1 - 13.8iT - 49T^{2} \) |
| 11 | \( 1 - 13.0iT - 121T^{2} \) |
| 13 | \( 1 + 7.16T + 169T^{2} \) |
| 17 | \( 1 - 31.5T + 289T^{2} \) |
| 19 | \( 1 - 16.4iT - 361T^{2} \) |
| 23 | \( 1 - 16.9iT - 529T^{2} \) |
| 29 | \( 1 - 42.4T + 841T^{2} \) |
| 31 | \( 1 - 29.6iT - 961T^{2} \) |
| 37 | \( 1 + 39.3T + 1.36e3T^{2} \) |
| 41 | \( 1 - 39.8T + 1.68e3T^{2} \) |
| 43 | \( 1 - 16.3iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 57.8iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 46.4T + 2.80e3T^{2} \) |
| 59 | \( 1 + 14.2iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 63.7T + 3.72e3T^{2} \) |
| 67 | \( 1 + 32.5iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 22.4iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 24.9T + 5.32e3T^{2} \) |
| 79 | \( 1 + 61.9iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 44.7iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 1.95T + 7.92e3T^{2} \) |
| 97 | \( 1 + 44.5T + 9.40e3T^{2} \) |
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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
−9.566861090195244830587280942842, −8.608941513033665279249545502581, −7.943128411736821598429057959524, −7.40217911629543310803399993743, −6.26925379998407604038525483325, −5.35700249757361263240589631898, −4.73139002349061859527212889794, −3.51879626674420204093894745036, −2.74195701318936667850682720914, −1.52592501858461000623439795519,
0.40164512519473449376161214319, 0.923979759052119839117876720913, 2.98429995071267744654173145806, 3.72904659937625251600358799773, 4.34819918599531561834357477066, 5.32961194293462331536804130855, 6.62476574326755147194043331256, 7.25185353118116026558533376511, 7.939367238433302250974553271882, 8.396616289833407213730855000415
