| L(s) = 1 | + 6.70·5-s + 11.6i·7-s + 15.5i·11-s + 8·13-s + 26.8·17-s + 23.2i·19-s − 31.1i·23-s + 20.0·25-s + 13.4·29-s − 11.6i·31-s + 77.9i·35-s − 2·37-s − 40.2·41-s + 23.2i·43-s − 31.1i·47-s + ⋯ |
| L(s) = 1 | + 1.34·5-s + 1.65i·7-s + 1.41i·11-s + 0.615·13-s + 1.57·17-s + 1.22i·19-s − 1.35i·23-s + 0.800·25-s + 0.462·29-s − 0.374i·31-s + 2.22i·35-s − 0.0540·37-s − 0.981·41-s + 0.540i·43-s − 0.663i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.821515203\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.821515203\) |
| \(L(2)\) |
|
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 | \( 1 - 6.70T + 25T^{2} \) |
| 7 | \( 1 - 11.6iT - 49T^{2} \) |
| 11 | \( 1 - 15.5iT - 121T^{2} \) |
| 13 | \( 1 - 8T + 169T^{2} \) |
| 17 | \( 1 - 26.8T + 289T^{2} \) |
| 19 | \( 1 - 23.2iT - 361T^{2} \) |
| 23 | \( 1 + 31.1iT - 529T^{2} \) |
| 29 | \( 1 - 13.4T + 841T^{2} \) |
| 31 | \( 1 + 11.6iT - 961T^{2} \) |
| 37 | \( 1 + 2T + 1.36e3T^{2} \) |
| 41 | \( 1 + 40.2T + 1.68e3T^{2} \) |
| 43 | \( 1 - 23.2iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 31.1iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 20.1T + 2.80e3T^{2} \) |
| 59 | \( 1 - 62.3iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 104T + 3.72e3T^{2} \) |
| 67 | \( 1 + 69.7iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 62.3iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 61T + 5.32e3T^{2} \) |
| 79 | \( 1 + 92.9iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 77.9iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 147.T + 7.92e3T^{2} \) |
| 97 | \( 1 - 103T + 9.40e3T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.459235532562199575587687134658, −8.630595363604033295840692567379, −7.922256485796657042047800042580, −6.72008435989752822138947917439, −5.91013711555408519292102559470, −5.53694175776632457968551906662, −4.57168334969646327102594620875, −3.14181837639446801885348312499, −2.18939311894684469141543399841, −1.53201120816612171372504517416,
0.77462535524432316909202571365, 1.48150001799807314259330827012, 3.07591883702162120478574104304, 3.68489600185974284038891204691, 4.98203935184785359111739086473, 5.74050288954074986026344651373, 6.46714516588272109471462504129, 7.31008008050310419884019290064, 8.122300814945045055992344565534, 9.066183820075632794947217448073
