L(s) = 1 | + (1.81 + 1.04i)2-s + (1.19 + 2.07i)4-s + (−1.04 − 1.80i)5-s + 0.819i·8-s − 4.37i·10-s + (2.79 + 1.61i)11-s + (2.68 − 1.55i)13-s + (1.53 − 2.65i)16-s − 1.63·17-s − 5.53i·19-s + (2.49 − 4.32i)20-s + (3.38 + 5.85i)22-s + (−1.00 + 0.580i)23-s + (0.316 − 0.547i)25-s + 6.50·26-s + ⋯ |
L(s) = 1 | + (1.28 + 0.740i)2-s + (0.597 + 1.03i)4-s + (−0.467 − 0.809i)5-s + 0.289i·8-s − 1.38i·10-s + (0.843 + 0.486i)11-s + (0.745 − 0.430i)13-s + (0.383 − 0.663i)16-s − 0.395·17-s − 1.26i·19-s + (0.558 − 0.967i)20-s + (0.721 + 1.24i)22-s + (−0.209 + 0.121i)23-s + (0.0632 − 0.109i)25-s + 1.27·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 - 0.0785i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.996 - 0.0785i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.267373206\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.267373206\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (-1.81 - 1.04i)T + (1 + 1.73i)T^{2} \) |
| 5 | \( 1 + (1.04 + 1.80i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-2.79 - 1.61i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-2.68 + 1.55i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + 1.63T + 17T^{2} \) |
| 19 | \( 1 + 5.53iT - 19T^{2} \) |
| 23 | \( 1 + (1.00 - 0.580i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-7.05 - 4.07i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-5.16 + 2.98i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 5.65T + 37T^{2} \) |
| 41 | \( 1 + (-1.35 - 2.34i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (0.974 - 1.68i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (4.06 - 7.04i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + 6.09iT - 53T^{2} \) |
| 59 | \( 1 + (-1.98 - 3.43i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-4.15 - 2.39i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-0.336 - 0.583i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 7.01iT - 71T^{2} \) |
| 73 | \( 1 + 3.42iT - 73T^{2} \) |
| 79 | \( 1 + (-7.07 + 12.2i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (1.54 - 2.67i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + 4.91T + 89T^{2} \) |
| 97 | \( 1 + (-2.07 - 1.20i)T + (48.5 + 84.0i)T^{2} \) |
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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
−9.455549919769501811350931212841, −8.641297129831003835460691554853, −7.903433129539760665926959257138, −6.79516794518888454594725267215, −6.40848629293470785919332669471, −5.23971278539531189086398200343, −4.60203217404072514320779701740, −3.96817620544200676373197101392, −2.86849345932315781563333484321, −1.02395926704506615790178479298,
1.50973412934107604393268675728, 2.75005588624086062968334985899, 3.66458741917810459928386466126, 4.09857548384214043508604061115, 5.25424266293697950720898851367, 6.28988686393633982847492659633, 6.72252523478299845784159123008, 8.076668371504532273953314776976, 8.725422538660232473241418006704, 10.00940874911730984300576254280