| L(s) = 1 | − 5.03·2-s − 8.47·3-s + 17.3·4-s − 0.885·5-s + 42.6·6-s − 3.81·7-s − 46.9·8-s + 44.8·9-s + 4.45·10-s + 52.3·11-s − 146.·12-s − 8.06·13-s + 19.2·14-s + 7.50·15-s + 97.5·16-s − 225.·18-s − 66.5·19-s − 15.3·20-s + 32.3·21-s − 263.·22-s − 180.·23-s + 397.·24-s − 124.·25-s + 40.5·26-s − 151.·27-s − 66.1·28-s + 41.2·29-s + ⋯ |
| L(s) = 1 | − 1.77·2-s − 1.63·3-s + 2.16·4-s − 0.0792·5-s + 2.90·6-s − 0.206·7-s − 2.07·8-s + 1.66·9-s + 0.140·10-s + 1.43·11-s − 3.53·12-s − 0.171·13-s + 0.366·14-s + 0.129·15-s + 1.52·16-s − 2.95·18-s − 0.803·19-s − 0.171·20-s + 0.336·21-s − 2.55·22-s − 1.63·23-s + 3.38·24-s − 0.993·25-s + 0.305·26-s − 1.07·27-s − 0.446·28-s + 0.264·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.2963821998\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2963821998\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 \) |
| good | 2 | \( 1 + 5.03T + 8T^{2} \) |
| 3 | \( 1 + 8.47T + 27T^{2} \) |
| 5 | \( 1 + 0.885T + 125T^{2} \) |
| 7 | \( 1 + 3.81T + 343T^{2} \) |
| 11 | \( 1 - 52.3T + 1.33e3T^{2} \) |
| 13 | \( 1 + 8.06T + 2.19e3T^{2} \) |
| 19 | \( 1 + 66.5T + 6.85e3T^{2} \) |
| 23 | \( 1 + 180.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 41.2T + 2.43e4T^{2} \) |
| 31 | \( 1 - 34.9T + 2.97e4T^{2} \) |
| 37 | \( 1 + 130.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 17.9T + 6.89e4T^{2} \) |
| 43 | \( 1 - 277.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 463.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 329.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 678.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 340.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 15.3T + 3.00e5T^{2} \) |
| 71 | \( 1 - 670.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 193.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.08e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 865.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.12e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 379.T + 9.12e5T^{2} \) |
| 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
−11.25343183485026191533442273684, −10.31251506988237688300320225745, −9.700804363694586511544341549430, −8.646066188835189082465595813330, −7.46273662908480002917341562695, −6.50489073890688111542814836182, −5.95133000359171548499051263504, −4.18155433044405949340996349645, −1.81519181062734399998270758839, −0.54061341279922133474278440778,
0.54061341279922133474278440778, 1.81519181062734399998270758839, 4.18155433044405949340996349645, 5.95133000359171548499051263504, 6.50489073890688111542814836182, 7.46273662908480002917341562695, 8.646066188835189082465595813330, 9.700804363694586511544341549430, 10.31251506988237688300320225745, 11.25343183485026191533442273684
