L(s) = 1 | + (0.5 + 0.866i)2-s + (−0.499 + 0.866i)4-s + (−0.5 + 0.866i)7-s − 0.999·8-s + (1.5 + 2.59i)11-s + (−3.5 − 0.866i)13-s − 0.999·14-s + (−0.5 − 0.866i)16-s + (1.5 − 2.59i)17-s + (−2.5 + 4.33i)19-s + (−1.5 + 2.59i)22-s + (3 + 5.19i)23-s − 5·25-s + (−1 − 3.46i)26-s + (−0.499 − 0.866i)28-s + (−1.5 − 2.59i)29-s + ⋯ |
L(s) = 1 | + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s + (−0.188 + 0.327i)7-s − 0.353·8-s + (0.452 + 0.783i)11-s + (−0.970 − 0.240i)13-s − 0.267·14-s + (−0.125 − 0.216i)16-s + (0.363 − 0.630i)17-s + (−0.573 + 0.993i)19-s + (−0.319 + 0.553i)22-s + (0.625 + 1.08i)23-s − 25-s + (−0.196 − 0.679i)26-s + (−0.0944 − 0.163i)28-s + (−0.278 − 0.482i)29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0256i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0256i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.002653584\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.002653584\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 + (0.5 - 0.866i)T \) |
| 13 | \( 1 + (3.5 + 0.866i)T \) |
good | 5 | \( 1 + 5T^{2} \) |
| 11 | \( 1 + (-1.5 - 2.59i)T + (-5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + (-1.5 + 2.59i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (2.5 - 4.33i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-3 - 5.19i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (1.5 + 2.59i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 + (-2 - 3.46i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (1.5 + 2.59i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (4 - 6.92i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + 9T + 47T^{2} \) |
| 53 | \( 1 - 9T + 53T^{2} \) |
| 59 | \( 1 + (3 - 5.19i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (2.5 - 4.33i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (7 + 12.1i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (3 - 5.19i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + 4T + 73T^{2} \) |
| 79 | \( 1 + T + 79T^{2} \) |
| 83 | \( 1 - 6T + 83T^{2} \) |
| 89 | \( 1 + (4.5 + 7.79i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (4 - 6.92i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−9.653442464396652947724529187769, −9.051297610560610383708346398818, −7.88667520343074473433397400528, −7.45813232339023579332972048810, −6.55542243702444089986673865799, −5.69922858977973432158621707153, −4.95931159244578837584615899882, −4.03128775332095571184509956250, −3.03596539766522234745644613513, −1.79733926639099007997162596308,
0.32319354833279571260236291351, 1.81307851726224342351687968094, 2.91722754221176858081016022074, 3.84153033207409697659694930803, 4.67141318112189569278912530795, 5.60276589977837162017963605570, 6.52135518519726336874058057142, 7.25860721786543634265575165528, 8.390590308237013648968201641780, 9.071683161936398029196703161524