L(s) = 1 | + (0.850 + 0.490i)2-s + (−0.518 − 0.897i)4-s + (0.940 + 1.62i)5-s − 2.98i·8-s + 1.84i·10-s + (−3.54 − 2.04i)11-s + (3.51 − 2.02i)13-s + (0.426 − 0.738i)16-s − 1.62·17-s − 8.12i·19-s + (0.974 − 1.68i)20-s + (−2.00 − 3.47i)22-s + (−3.73 + 2.15i)23-s + (0.730 − 1.26i)25-s + 3.98·26-s + ⋯ |
L(s) = 1 | + (0.601 + 0.347i)2-s + (−0.259 − 0.448i)4-s + (0.420 + 0.728i)5-s − 1.05i·8-s + 0.583i·10-s + (−1.06 − 0.616i)11-s + (0.974 − 0.562i)13-s + (0.106 − 0.184i)16-s − 0.393·17-s − 1.86i·19-s + (0.217 − 0.377i)20-s + (−0.427 − 0.740i)22-s + (−0.778 + 0.449i)23-s + (0.146 − 0.253i)25-s + 0.781·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.456 + 0.889i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.456 + 0.889i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.861319873\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.861319873\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-0.850 - 0.490i)T + (1 + 1.73i)T^{2} \) |
| 5 | \( 1 + (-0.940 - 1.62i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (3.54 + 2.04i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-3.51 + 2.02i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + 1.62T + 17T^{2} \) |
| 19 | \( 1 + 8.12iT - 19T^{2} \) |
| 23 | \( 1 + (3.73 - 2.15i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-0.542 - 0.313i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-3.69 + 2.13i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 7.94T + 37T^{2} \) |
| 41 | \( 1 + (0.912 + 1.57i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (3.53 - 6.12i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-3.96 + 6.87i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + 8.37iT - 53T^{2} \) |
| 59 | \( 1 + (-4.08 - 7.07i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-3.24 - 1.87i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (6.26 + 10.8i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 14.4iT - 71T^{2} \) |
| 73 | \( 1 + 3.78iT - 73T^{2} \) |
| 79 | \( 1 + (4.18 - 7.24i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (4.38 - 7.59i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 - 9.80T + 89T^{2} \) |
| 97 | \( 1 + (-11.4 - 6.61i)T + (48.5 + 84.0i)T^{2} \) |
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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
−9.574480109996616671624514723269, −8.679867974430608025212508551090, −7.77426735024088981273593004990, −6.68726887006373945158380578894, −6.15767623428556399700041257026, −5.37502774057023896143205398816, −4.53132128886367088360417419308, −3.36807249314810687523465395873, −2.43657280366187950807015411405, −0.64008932620281021315592230892,
1.59918747077457536338050635315, 2.65455749229297593965867839191, 3.88947997666585239716931506641, 4.52848628362361331691392291397, 5.45344240017695379588393380309, 6.17104068699011591186105336115, 7.51666709212952219248077556157, 8.272094281150335131601343155450, 8.818440510601184427560014742056, 9.841693849445907805326183373612