L(s) = 1 | + (1.18 + 0.429i)2-s + (−0.523 − 2.96i)3-s + (−0.321 − 0.270i)4-s + (0.657 − 3.72i)6-s + (1.86 − 3.22i)7-s + (−1.52 − 2.63i)8-s + (−5.70 + 2.07i)9-s + (1.67 + 2.90i)11-s + (−0.632 + 1.09i)12-s + (−0.840 + 4.76i)13-s + (3.58 − 3.00i)14-s + (−0.518 − 2.93i)16-s + (−2.51 − 0.914i)17-s − 7.63·18-s + (0.961 − 4.25i)19-s + ⋯ |
L(s) = 1 | + (0.835 + 0.303i)2-s + (−0.301 − 1.71i)3-s + (−0.160 − 0.135i)4-s + (0.268 − 1.52i)6-s + (0.703 − 1.21i)7-s + (−0.537 − 0.931i)8-s + (−1.90 + 0.692i)9-s + (0.505 + 0.876i)11-s + (−0.182 + 0.316i)12-s + (−0.233 + 1.32i)13-s + (0.958 − 0.804i)14-s + (−0.129 − 0.734i)16-s + (−0.609 − 0.221i)17-s − 1.79·18-s + (0.220 − 0.975i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.706 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.706 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.630640 - 1.52177i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.630640 - 1.52177i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 19 | \( 1 + (-0.961 + 4.25i)T \) |
good | 2 | \( 1 + (-1.18 - 0.429i)T + (1.53 + 1.28i)T^{2} \) |
| 3 | \( 1 + (0.523 + 2.96i)T + (-2.81 + 1.02i)T^{2} \) |
| 7 | \( 1 + (-1.86 + 3.22i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-1.67 - 2.90i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (0.840 - 4.76i)T + (-12.2 - 4.44i)T^{2} \) |
| 17 | \( 1 + (2.51 + 0.914i)T + (13.0 + 10.9i)T^{2} \) |
| 23 | \( 1 + (1.43 + 1.20i)T + (3.99 + 22.6i)T^{2} \) |
| 29 | \( 1 + (-4.93 + 1.79i)T + (22.2 - 18.6i)T^{2} \) |
| 31 | \( 1 + (-1.55 + 2.70i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 0.992T + 37T^{2} \) |
| 41 | \( 1 + (-0.0723 - 0.410i)T + (-38.5 + 14.0i)T^{2} \) |
| 43 | \( 1 + (-5.52 + 4.64i)T + (7.46 - 42.3i)T^{2} \) |
| 47 | \( 1 + (-2.10 + 0.766i)T + (36.0 - 30.2i)T^{2} \) |
| 53 | \( 1 + (0.199 + 0.167i)T + (9.20 + 52.1i)T^{2} \) |
| 59 | \( 1 + (-4.87 - 1.77i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (-0.589 - 0.494i)T + (10.5 + 60.0i)T^{2} \) |
| 67 | \( 1 + (-10.1 + 3.68i)T + (51.3 - 43.0i)T^{2} \) |
| 71 | \( 1 + (-1.53 + 1.28i)T + (12.3 - 69.9i)T^{2} \) |
| 73 | \( 1 + (0.792 + 4.49i)T + (-68.5 + 24.9i)T^{2} \) |
| 79 | \( 1 + (2.09 + 11.8i)T + (-74.2 + 27.0i)T^{2} \) |
| 83 | \( 1 + (6.78 - 11.7i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (1.33 - 7.55i)T + (-83.6 - 30.4i)T^{2} \) |
| 97 | \( 1 + (6.79 + 2.47i)T + (74.3 + 62.3i)T^{2} \) |
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
−11.07383345247217529045776281644, −9.761903184003163085011335172334, −8.662015108003607620262504041090, −7.35646949944095327994857836698, −6.95710762369598687880757505673, −6.25696245216060028862931590253, −4.86511613768710504851050406911, −4.17630871112662267388890870675, −2.15611299589223434089874806721, −0.841655511667984053854185386642,
2.76425044304747317668223343897, 3.63719255636572424045204073610, 4.62595491886341719014034422035, 5.47036653635094207828144505409, 5.90350020667524024170923758582, 8.290847316971069085613482420274, 8.630047489693172888281642559922, 9.650497258510142501444648677912, 10.64255241582270538407276136752, 11.39688620519593334484290668939