| L(s) = 1 | − 1.52·2-s − 5.67·4-s + 9.65·5-s − 22.3·7-s + 20.8·8-s − 14.7·10-s − 50.3·11-s + 34.0·14-s + 13.6·16-s − 86.1·17-s − 116.·19-s − 54.8·20-s + 76.6·22-s − 72·23-s − 31.7·25-s + 126.·28-s − 14.1·29-s + 196.·31-s − 187.·32-s + 131.·34-s − 216·35-s + 154.·37-s + 178.·38-s + 201.·40-s − 265.·41-s + 211.·43-s + 285.·44-s + ⋯ |
| L(s) = 1 | − 0.538·2-s − 0.709·4-s + 0.863·5-s − 1.20·7-s + 0.921·8-s − 0.465·10-s − 1.37·11-s + 0.650·14-s + 0.213·16-s − 1.22·17-s − 1.41·19-s − 0.613·20-s + 0.742·22-s − 0.652·23-s − 0.253·25-s + 0.857·28-s − 0.0905·29-s + 1.13·31-s − 1.03·32-s + 0.662·34-s − 1.04·35-s + 0.686·37-s + 0.760·38-s + 0.795·40-s − 1.01·41-s + 0.751·43-s + 0.978·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.3597956069\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3597956069\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 1.52T + 8T^{2} \) |
| 5 | \( 1 - 9.65T + 125T^{2} \) |
| 7 | \( 1 + 22.3T + 343T^{2} \) |
| 11 | \( 1 + 50.3T + 1.33e3T^{2} \) |
| 17 | \( 1 + 86.1T + 4.91e3T^{2} \) |
| 19 | \( 1 + 116.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 72T + 1.21e4T^{2} \) |
| 29 | \( 1 + 14.1T + 2.43e4T^{2} \) |
| 31 | \( 1 - 196.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 154.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 265.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 211.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 67.5T + 1.03e5T^{2} \) |
| 53 | \( 1 + 686.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 91.9T + 2.05e5T^{2} \) |
| 61 | \( 1 - 329.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 768.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 264.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 771.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.22e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 514.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 527.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 74.2T + 9.12e5T^{2} \) |
| 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
−9.188629271165100425861958656697, −8.449527538391811575833583209922, −7.70402601297001074660193297164, −6.53627898323159743360614431951, −5.99338064002945197627478154690, −4.92789887017885145648806710783, −4.11207453878641751891100709910, −2.81321100793184268722843225666, −1.92855312603084206193136482107, −0.30225270113315846921080493552,
0.30225270113315846921080493552, 1.92855312603084206193136482107, 2.81321100793184268722843225666, 4.11207453878641751891100709910, 4.92789887017885145648806710783, 5.99338064002945197627478154690, 6.53627898323159743360614431951, 7.70402601297001074660193297164, 8.449527538391811575833583209922, 9.188629271165100425861958656697
