L(s) = 1 | + (−0.5 + 0.866i)2-s + (−0.499 − 0.866i)4-s + (2.61 + 0.397i)7-s + 0.999·8-s + (0.429 − 0.248i)11-s − 2.74·13-s + (−1.65 + 2.06i)14-s + (−0.5 + 0.866i)16-s + (3.16 − 1.82i)17-s + (−3.12 − 1.80i)19-s + 0.496i·22-s + (−3.21 + 5.56i)23-s + (1.37 − 2.37i)26-s + (−0.963 − 2.46i)28-s + 8.87i·29-s + ⋯ |
L(s) = 1 | + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s + (0.988 + 0.150i)7-s + 0.353·8-s + (0.129 − 0.0748i)11-s − 0.761·13-s + (−0.441 + 0.552i)14-s + (−0.125 + 0.216i)16-s + (0.768 − 0.443i)17-s + (−0.716 − 0.413i)19-s + 0.105i·22-s + (−0.669 + 1.16i)23-s + (0.269 − 0.466i)26-s + (−0.182 − 0.465i)28-s + 1.64i·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.579 - 0.815i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.579 - 0.815i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.172850239\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.172850239\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 - 0.866i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-2.61 - 0.397i)T \) |
good | 11 | \( 1 + (-0.429 + 0.248i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + 2.74T + 13T^{2} \) |
| 17 | \( 1 + (-3.16 + 1.82i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (3.12 + 1.80i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (3.21 - 5.56i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 8.87iT - 29T^{2} \) |
| 31 | \( 1 + (6.90 - 3.98i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (1.98 + 1.14i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 2.22T + 41T^{2} \) |
| 43 | \( 1 + 2.22iT - 43T^{2} \) |
| 47 | \( 1 + (-5.66 - 3.27i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-3.88 - 6.72i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-3.05 - 5.28i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-3.24 - 1.87i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-7.08 + 4.08i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 10.3iT - 71T^{2} \) |
| 73 | \( 1 + (6.53 + 11.3i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (4.44 - 7.69i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 4.79iT - 83T^{2} \) |
| 89 | \( 1 + (0.743 - 1.28i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 9.05T + 97T^{2} \) |
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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
−8.873847766703315298545815355262, −8.185268288888738195421116396932, −7.29410679870162252033707414634, −7.07982373908708093963910328519, −5.73701457070554383448548387707, −5.31357349921748541782527700442, −4.51294193437958122552198139655, −3.47479243070455586117822223866, −2.20143866198516728037186537988, −1.21628848267665165859492623707,
0.42942452162149267755133422479, 1.81693232569039083849698049398, 2.39766781930435472219514415862, 3.76265465112195692799344968406, 4.33160608890495408077788328544, 5.24512176733257976721971577642, 6.11364005756875293992082990570, 7.14899420074349837222187148618, 7.933031437949629232356087958660, 8.325617221980987450117899696602