| L(s) = 1 | + (0.448 − 0.497i)2-s + (2.86 − 1.27i)3-s + (0.162 + 1.54i)4-s + (−2.09 + 0.445i)5-s + (0.649 − 1.99i)6-s + (1.92 + 1.39i)8-s + (4.58 − 5.09i)9-s + (−0.717 + 1.24i)10-s + (3.25 + 0.628i)11-s + (2.43 + 4.21i)12-s + (0.781 + 2.40i)13-s + (−5.44 + 3.95i)15-s + (−1.47 + 0.313i)16-s + (−1.19 − 1.33i)17-s + (−0.480 − 4.56i)18-s + (0.703 − 6.69i)19-s + ⋯ |
| L(s) = 1 | + (0.316 − 0.351i)2-s + (1.65 − 0.737i)3-s + (0.0810 + 0.771i)4-s + (−0.937 + 0.199i)5-s + (0.265 − 0.816i)6-s + (0.680 + 0.494i)8-s + (1.52 − 1.69i)9-s + (−0.227 + 0.393i)10-s + (0.981 + 0.189i)11-s + (0.703 + 1.21i)12-s + (0.216 + 0.666i)13-s + (−1.40 + 1.02i)15-s + (−0.369 + 0.0784i)16-s + (−0.290 − 0.322i)17-s + (−0.113 − 1.07i)18-s + (0.161 − 1.53i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.911 + 0.411i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.911 + 0.411i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.64202 - 0.568199i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.64202 - 0.568199i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 11 | \( 1 + (-3.25 - 0.628i)T \) |
| good | 2 | \( 1 + (-0.448 + 0.497i)T + (-0.209 - 1.98i)T^{2} \) |
| 3 | \( 1 + (-2.86 + 1.27i)T + (2.00 - 2.22i)T^{2} \) |
| 5 | \( 1 + (2.09 - 0.445i)T + (4.56 - 2.03i)T^{2} \) |
| 13 | \( 1 + (-0.781 - 2.40i)T + (-10.5 + 7.64i)T^{2} \) |
| 17 | \( 1 + (1.19 + 1.33i)T + (-1.77 + 16.9i)T^{2} \) |
| 19 | \( 1 + (-0.703 + 6.69i)T + (-18.5 - 3.95i)T^{2} \) |
| 23 | \( 1 + (-1.58 - 2.74i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-0.747 + 0.543i)T + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (2.93 + 0.624i)T + (28.3 + 12.6i)T^{2} \) |
| 37 | \( 1 + (1.37 + 0.613i)T + (24.7 + 27.4i)T^{2} \) |
| 41 | \( 1 + (4.49 + 3.26i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + 8.42T + 43T^{2} \) |
| 47 | \( 1 + (0.459 - 4.37i)T + (-45.9 - 9.77i)T^{2} \) |
| 53 | \( 1 + (0.652 + 0.138i)T + (48.4 + 21.5i)T^{2} \) |
| 59 | \( 1 + (-0.0385 - 0.366i)T + (-57.7 + 12.2i)T^{2} \) |
| 61 | \( 1 + (4.90 - 1.04i)T + (55.7 - 24.8i)T^{2} \) |
| 67 | \( 1 + (-0.451 + 0.781i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (4.59 - 14.1i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (0.840 + 7.99i)T + (-71.4 + 15.1i)T^{2} \) |
| 79 | \( 1 + (-2.71 + 3.01i)T + (-8.25 - 78.5i)T^{2} \) |
| 83 | \( 1 + (1.25 - 3.85i)T + (-67.1 - 48.7i)T^{2} \) |
| 89 | \( 1 + (-4.15 - 7.19i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-2.63 - 8.09i)T + (-78.4 + 57.0i)T^{2} \) |
| 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
−11.16585331261293993662946033295, −9.456590743488329304878400841842, −8.849103272859573908480345573913, −8.107642597209567037092796017397, −7.14119692035164259110912583772, −6.90759232518335328377803406914, −4.49238927503320996739354864967, −3.68539111345569506459767777503, −2.95405215586104184411138098233, −1.77421774176279048575968266525,
1.68228235154610373598169409904, 3.39290268614565612271900514701, 4.01484316121663469413412539722, 4.97448710640903154733554568067, 6.34282289138252700376448226467, 7.52424352918975392425892788103, 8.287382800461282888389867000585, 8.980548117533406979811859953409, 9.981580824510661995460668722399, 10.52534472109192668306550810294
