L(s) = 1 | + (1.64 − 0.545i)3-s + (−0.794 − 1.37i)5-s + (−1.23 + 2.33i)7-s + (2.40 − 1.79i)9-s + (−0.794 + 1.37i)11-s + (2.40 − 4.16i)13-s + (−2.05 − 1.82i)15-s + (−2.69 − 4.67i)17-s + (3.54 − 6.14i)19-s + (−0.761 + 4.51i)21-s + (0.150 + 0.260i)23-s + (1.23 − 2.14i)25-s + (2.97 − 4.25i)27-s + (4.13 + 7.16i)29-s + 2.71·31-s + ⋯ |
L(s) = 1 | + (0.949 − 0.314i)3-s + (−0.355 − 0.615i)5-s + (−0.468 + 0.883i)7-s + (0.801 − 0.597i)9-s + (−0.239 + 0.414i)11-s + (0.667 − 1.15i)13-s + (−0.530 − 0.472i)15-s + (−0.654 − 1.13i)17-s + (0.814 − 1.41i)19-s + (−0.166 + 0.986i)21-s + (0.0313 + 0.0542i)23-s + (0.247 − 0.429i)25-s + (0.572 − 0.819i)27-s + (0.768 + 1.33i)29-s + 0.487·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.381 + 0.924i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.381 + 0.924i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.971908666\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.971908666\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.64 + 0.545i)T \) |
| 7 | \( 1 + (1.23 - 2.33i)T \) |
good | 5 | \( 1 + (0.794 + 1.37i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (0.794 - 1.37i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-2.40 + 4.16i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (2.69 + 4.67i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-3.54 + 6.14i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.150 - 0.260i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-4.13 - 7.16i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 2.71T + 31T^{2} \) |
| 37 | \( 1 + (-0.5 + 0.866i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-2.93 + 5.08i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-0.833 - 1.44i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + 2.66T + 47T^{2} \) |
| 53 | \( 1 + (-2.44 - 4.23i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + 6.47T + 59T^{2} \) |
| 61 | \( 1 + 4.47T + 61T^{2} \) |
| 67 | \( 1 - 10.0T + 67T^{2} \) |
| 71 | \( 1 + 12.7T + 71T^{2} \) |
| 73 | \( 1 + (-8.02 - 13.9i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + 8.38T + 79T^{2} \) |
| 83 | \( 1 + (1.18 + 2.04i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-1.60 + 2.78i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-0.712 - 1.23i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−9.476864462198204342438253484012, −8.921951959289994230845196693530, −8.333884178066667171072430572951, −7.37955748933619450663014303786, −6.61837400366382200691221802254, −5.35813398189847867956567741615, −4.53735514978530019890637063504, −3.14099252249708479365203334680, −2.57490836710793846027632685661, −0.867995735331923963404615627086,
1.58519145803084804175052005472, 3.02028627755577217815547505454, 3.80587243408054002686252434227, 4.42408101901510414349463888895, 6.08797020147025269176192099419, 6.81782502504492113906862851829, 7.78182592752254096082298172937, 8.339999466315244440137123589916, 9.363145636115573424240018203325, 10.09964724662244201822311452519