| L(s) = 1 | + 0.322i·3-s + (−0.824 − 2.53i)5-s + (2.33 + 3.21i)7-s + 2.89·9-s + (−1.25 − 0.408i)11-s + (−0.327 + 0.450i)13-s + (0.818 − 0.265i)15-s + (1.71 + 0.557i)17-s + (−3.03 − 4.18i)19-s + (−1.03 + 0.754i)21-s + (6.14 + 4.46i)23-s + (−1.71 + 1.24i)25-s + 1.90i·27-s + (5.02 − 1.63i)29-s + (2.37 − 7.32i)31-s + ⋯ |
| L(s) = 1 | + 0.186i·3-s + (−0.368 − 1.13i)5-s + (0.884 + 1.21i)7-s + 0.965·9-s + (−0.378 − 0.123i)11-s + (−0.0908 + 0.125i)13-s + (0.211 − 0.0686i)15-s + (0.416 + 0.135i)17-s + (−0.697 − 0.959i)19-s + (−0.226 + 0.164i)21-s + (1.28 + 0.930i)23-s + (−0.342 + 0.249i)25-s + 0.365i·27-s + (0.932 − 0.302i)29-s + (0.427 − 1.31i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 656 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0633i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 656 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 + 0.0633i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.64523 - 0.0521998i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.64523 - 0.0521998i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 41 | \( 1 + (-2.75 + 5.78i)T \) |
| good | 3 | \( 1 - 0.322iT - 3T^{2} \) |
| 5 | \( 1 + (0.824 + 2.53i)T + (-4.04 + 2.93i)T^{2} \) |
| 7 | \( 1 + (-2.33 - 3.21i)T + (-2.16 + 6.65i)T^{2} \) |
| 11 | \( 1 + (1.25 + 0.408i)T + (8.89 + 6.46i)T^{2} \) |
| 13 | \( 1 + (0.327 - 0.450i)T + (-4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (-1.71 - 0.557i)T + (13.7 + 9.99i)T^{2} \) |
| 19 | \( 1 + (3.03 + 4.18i)T + (-5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 + (-6.14 - 4.46i)T + (7.10 + 21.8i)T^{2} \) |
| 29 | \( 1 + (-5.02 + 1.63i)T + (23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (-2.37 + 7.32i)T + (-25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (-3.54 - 10.9i)T + (-29.9 + 21.7i)T^{2} \) |
| 43 | \( 1 + (0.0631 + 0.0458i)T + (13.2 + 40.8i)T^{2} \) |
| 47 | \( 1 + (-1.97 + 2.71i)T + (-14.5 - 44.6i)T^{2} \) |
| 53 | \( 1 + (-8.61 + 2.79i)T + (42.8 - 31.1i)T^{2} \) |
| 59 | \( 1 + (-1.68 - 1.22i)T + (18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (5.11 - 3.71i)T + (18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + (8.29 - 2.69i)T + (54.2 - 39.3i)T^{2} \) |
| 71 | \( 1 + (9.61 + 3.12i)T + (57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + 9.63T + 73T^{2} \) |
| 79 | \( 1 - 10.8iT - 79T^{2} \) |
| 83 | \( 1 + 0.977T + 83T^{2} \) |
| 89 | \( 1 + (7.70 + 10.6i)T + (-27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (9.19 - 2.98i)T + (78.4 - 57.0i)T^{2} \) |
| 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
−10.52661543794474219692976467154, −9.478648847621341907104272531760, −8.743887023288430570355285693811, −8.150716429637919241858116981727, −7.13819148074352321017251029839, −5.76365120966072011408124317011, −4.88768336307418104621455842742, −4.33787547970405409555888850199, −2.62748326666741076866664685025, −1.22316430787157760853922300022,
1.24469776008030190553320084670, 2.81850893481144256315450799145, 4.03932712606335375016634235745, 4.80725788712071323006072328776, 6.34064622521189147848719334314, 7.29598940018368583882559600528, 7.52976058312427435751318002593, 8.640813625219325030796691987455, 10.13211067991886908508850707180, 10.56199909293295294546183157030
