L(s) = 1 | + (0.866 + 0.5i)2-s + (−0.5 + 0.866i)3-s + (0.499 + 0.866i)4-s + (−0.866 + 0.499i)6-s + (0.807 − 0.465i)7-s + 0.999i·8-s + (−0.499 − 0.866i)9-s + (1.45 + 0.841i)11-s − 0.999·12-s + (1.86 − 3.08i)13-s + 0.931·14-s + (−0.5 + 0.866i)16-s + (1.27 + 2.21i)17-s − 0.999i·18-s + (2.27 − 1.31i)19-s + ⋯ |
L(s) = 1 | + (0.612 + 0.353i)2-s + (−0.288 + 0.499i)3-s + (0.249 + 0.433i)4-s + (−0.353 + 0.204i)6-s + (0.305 − 0.176i)7-s + 0.353i·8-s + (−0.166 − 0.288i)9-s + (0.439 + 0.253i)11-s − 0.288·12-s + (0.516 − 0.856i)13-s + 0.249·14-s + (−0.125 + 0.216i)16-s + (0.309 + 0.536i)17-s − 0.235i·18-s + (0.522 − 0.301i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.505 - 0.862i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.505 - 0.862i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.567462271\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.567462271\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.866 - 0.5i)T \) |
| 3 | \( 1 + (0.5 - 0.866i)T \) |
| 5 | \( 1 \) |
| 13 | \( 1 + (-1.86 + 3.08i)T \) |
good | 7 | \( 1 + (-0.807 + 0.465i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-1.45 - 0.841i)T + (5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + (-1.27 - 2.21i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.27 + 1.31i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-3.22 + 5.59i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-1.69 + 2.93i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 - 9.29iT - 31T^{2} \) |
| 37 | \( 1 + (-7.93 - 4.57i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-1.01 - 0.587i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (1.47 + 2.55i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + 3.97iT - 47T^{2} \) |
| 53 | \( 1 + 5.36T + 53T^{2} \) |
| 59 | \( 1 + (3.44 - 1.98i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-4.36 - 7.55i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-4.49 - 2.59i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-3.40 + 1.96i)T + (35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + 7.26iT - 73T^{2} \) |
| 79 | \( 1 + 3.68T + 79T^{2} \) |
| 83 | \( 1 + 3.84iT - 83T^{2} \) |
| 89 | \( 1 + (-11.6 - 6.73i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-12.1 + 7.04i)T + (48.5 - 84.0i)T^{2} \) |
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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
−9.242753032112663141945651500324, −8.438517323358282152218110199898, −7.74565450161725971119417506672, −6.72355768152238189771014293973, −6.12248485724357732137312346827, −5.15880772720175633822756967597, −4.57013037537914071800069055953, −3.62107009830625798373906178766, −2.76761927562030608941989602013, −1.12426598978256441994374222072,
1.02097769590787542042763719834, 1.99718990694510341553815054844, 3.16571397386776838900763973304, 4.08860963960692145157645453428, 5.05676587249203855476292902787, 5.82415339687378933137691840910, 6.53335222478951148249081127494, 7.39212358849537867323630184177, 8.141433078057995524295263896753, 9.284058058712416629387238632858