| L(s) = 1 | + (0.467 − 1.35i)2-s + (−0.458 − 0.888i)3-s + (−0.0329 − 0.0259i)4-s + (−3.87 + 0.369i)5-s + (−1.41 + 0.203i)6-s + (−0.818 + 2.51i)7-s + (2.35 − 1.51i)8-s + (−0.580 + 0.814i)9-s + (−1.31 + 5.40i)10-s + (−2.90 + 1.00i)11-s + (−0.00793 + 0.0411i)12-s + (1.82 + 6.21i)13-s + (3.01 + 2.28i)14-s + (2.10 + 3.27i)15-s + (−0.962 − 3.96i)16-s + (−5.39 − 2.15i)17-s + ⋯ |
| L(s) = 1 | + (0.330 − 0.954i)2-s + (−0.264 − 0.513i)3-s + (−0.0164 − 0.0129i)4-s + (−1.73 + 0.165i)5-s + (−0.577 + 0.0830i)6-s + (−0.309 + 0.950i)7-s + (0.832 − 0.534i)8-s + (−0.193 + 0.271i)9-s + (−0.414 + 1.70i)10-s + (−0.876 + 0.303i)11-s + (−0.00229 + 0.0118i)12-s + (0.505 + 1.72i)13-s + (0.805 + 0.609i)14-s + (0.543 + 0.845i)15-s + (−0.240 − 0.991i)16-s + (−1.30 − 0.523i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.481 - 0.876i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.481 - 0.876i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.518750 + 0.306706i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.518750 + 0.306706i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.458 + 0.888i)T \) |
| 7 | \( 1 + (0.818 - 2.51i)T \) |
| 23 | \( 1 + (-1.31 - 4.61i)T \) |
| good | 2 | \( 1 + (-0.467 + 1.35i)T + (-1.57 - 1.23i)T^{2} \) |
| 5 | \( 1 + (3.87 - 0.369i)T + (4.90 - 0.946i)T^{2} \) |
| 11 | \( 1 + (2.90 - 1.00i)T + (8.64 - 6.79i)T^{2} \) |
| 13 | \( 1 + (-1.82 - 6.21i)T + (-10.9 + 7.02i)T^{2} \) |
| 17 | \( 1 + (5.39 + 2.15i)T + (12.3 + 11.7i)T^{2} \) |
| 19 | \( 1 + (-2.04 + 0.819i)T + (13.7 - 13.1i)T^{2} \) |
| 29 | \( 1 + (-0.917 - 6.38i)T + (-27.8 + 8.17i)T^{2} \) |
| 31 | \( 1 + (9.29 - 0.442i)T + (30.8 - 2.94i)T^{2} \) |
| 37 | \( 1 + (5.57 + 3.96i)T + (12.1 + 34.9i)T^{2} \) |
| 41 | \( 1 + (-0.0145 - 0.00663i)T + (26.8 + 30.9i)T^{2} \) |
| 43 | \( 1 + (-1.64 + 2.56i)T + (-17.8 - 39.1i)T^{2} \) |
| 47 | \( 1 + (3.20 - 1.85i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (1.78 + 1.87i)T + (-2.52 + 52.9i)T^{2} \) |
| 59 | \( 1 + (0.821 + 0.199i)T + (52.4 + 27.0i)T^{2} \) |
| 61 | \( 1 + (-6.26 - 3.23i)T + (35.3 + 49.6i)T^{2} \) |
| 67 | \( 1 + (-0.235 - 1.22i)T + (-62.2 + 24.9i)T^{2} \) |
| 71 | \( 1 + (-1.17 - 1.35i)T + (-10.1 + 70.2i)T^{2} \) |
| 73 | \( 1 + (1.50 - 1.91i)T + (-17.2 - 70.9i)T^{2} \) |
| 79 | \( 1 + (10.5 - 11.0i)T + (-3.75 - 78.9i)T^{2} \) |
| 83 | \( 1 + (-0.229 - 0.502i)T + (-54.3 + 62.7i)T^{2} \) |
| 89 | \( 1 + (0.376 - 7.90i)T + (-88.5 - 8.45i)T^{2} \) |
| 97 | \( 1 + (1.60 - 3.50i)T + (-63.5 - 73.3i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.27195748759274903425823873486, −10.94002967478718596286700715042, −9.327487725060606333788088149163, −8.475665811825993517921143534867, −7.15818051612462505995352100283, −6.99912825112562251972599961193, −5.17182625781909227665028431960, −4.10456371122198942512385964960, −3.12907083416560823447975912529, −1.93919564458838263201491268173,
0.32699724796012134577727293530, 3.27391597963289787878958312810, 4.23319671687445051823193336827, 5.08362534715877011594277169351, 6.19172359162595274456886667906, 7.25543636562923207489975371948, 7.929211134793409495910474869372, 8.553761648413202509494115913335, 10.35019817365535013771588237105, 10.78860894493311386193142177005
