L(s) = 1 | + (0.105 − 0.0611i)2-s + (−0.992 + 1.71i)4-s + 0.529·5-s + 0.487i·8-s + (0.0560 − 0.0323i)10-s − 4.20i·11-s + (1.74 − 1.00i)13-s + (−1.95 − 3.38i)16-s + (−2.19 − 3.79i)17-s + (4.54 + 2.62i)19-s + (−0.525 + 0.910i)20-s + (−0.257 − 0.445i)22-s − 6.27i·23-s − 4.71·25-s + (0.123 − 0.213i)26-s + ⋯ |
L(s) = 1 | + (0.0749 − 0.0432i)2-s + (−0.496 + 0.859i)4-s + 0.236·5-s + 0.172i·8-s + (0.0177 − 0.0102i)10-s − 1.26i·11-s + (0.484 − 0.279i)13-s + (−0.488 − 0.846i)16-s + (−0.532 − 0.921i)17-s + (1.04 + 0.601i)19-s + (−0.117 + 0.203i)20-s + (−0.0548 − 0.0949i)22-s − 1.30i·23-s − 0.943·25-s + (0.0242 − 0.0419i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.870 + 0.492i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.870 + 0.492i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.500149231\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.500149231\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-0.105 + 0.0611i)T + (1 - 1.73i)T^{2} \) |
| 5 | \( 1 - 0.529T + 5T^{2} \) |
| 11 | \( 1 + 4.20iT - 11T^{2} \) |
| 13 | \( 1 + (-1.74 + 1.00i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (2.19 + 3.79i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-4.54 - 2.62i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + 6.27iT - 23T^{2} \) |
| 29 | \( 1 + (-7.27 - 4.20i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-1.03 - 0.595i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-1.61 + 2.79i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (0.0994 + 0.172i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-3.96 + 6.86i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-4.98 - 8.63i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-3.65 + 2.10i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (6.71 - 11.6i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-11.3 + 6.55i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-3.29 + 5.69i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 8.50iT - 71T^{2} \) |
| 73 | \( 1 + (4.86 - 2.80i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (0.286 + 0.495i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-5.42 + 9.39i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (6.43 - 11.1i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (0.493 + 0.285i)T + (48.5 + 84.0i)T^{2} \) |
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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
−9.358290864998597274604064495586, −8.716952993918190934115749043755, −8.090208802901668329183248652740, −7.22894802881273097379093598176, −6.19777759671028689553456956364, −5.34084434742805846602370793270, −4.34450781137021688970678238834, −3.38099515280966366615467320257, −2.60483235664404071772779682599, −0.71601084265974836522480557593,
1.21538941324652870503717038572, 2.27162270461489514727920483010, 3.86354335738326055578081509443, 4.62700491941431900067162023973, 5.51530579276694513599456226763, 6.30115582647280076204374655347, 7.13644384990391548894044334520, 8.147412176558733215927785427825, 9.085417728709021940235897473102, 9.800643072293369874687856032103