L(s) = 1 | + (−0.105 − 0.0611i)2-s + (−0.992 − 1.71i)4-s + (0.264 + 0.458i)5-s + 0.487i·8-s − 0.0647i·10-s + (3.64 + 2.10i)11-s + (−1.74 + 1.00i)13-s + (−1.95 + 3.38i)16-s − 4.38·17-s + 5.24i·19-s + (0.525 − 0.910i)20-s + (−0.257 − 0.445i)22-s + (−5.43 + 3.13i)23-s + (2.35 − 4.08i)25-s + 0.246·26-s + ⋯ |
L(s) = 1 | + (−0.0749 − 0.0432i)2-s + (−0.496 − 0.859i)4-s + (0.118 + 0.205i)5-s + 0.172i·8-s − 0.0204i·10-s + (1.09 + 0.633i)11-s + (−0.484 + 0.279i)13-s + (−0.488 + 0.846i)16-s − 1.06·17-s + 1.20i·19-s + (0.117 − 0.203i)20-s + (−0.0548 − 0.0949i)22-s + (−1.13 + 0.654i)23-s + (0.471 − 0.817i)25-s + 0.0484·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.520 - 0.853i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.520 - 0.853i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.125228498\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.125228498\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-0.264 - 0.458i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-3.64 - 2.10i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (1.74 - 1.00i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + 4.38T + 17T^{2} \) |
| 19 | \( 1 - 5.24iT - 19T^{2} \) |
| 23 | \( 1 + (5.43 - 3.13i)T + (11.5 - 19.9i)T^{2} \) |
| 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 | \( 1 + 3.23T + 37T^{2} \) |
| 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 | \( 1 - 4.21iT - 53T^{2} \) |
| 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 | \( 1 - 5.61iT - 73T^{2} \) |
| 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 | \( 1 + 12.8T + 89T^{2} \) |
| 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
−9.964051663921939178694205488661, −8.996458688300204852169206740681, −8.403013532066410923190420674903, −7.12855163388891093244980008005, −6.44723134674943183723967403063, −5.65954263804140724665035677854, −4.55193836966388623146635332657, −3.98070501005085952798437784014, −2.33883511495289288689378266916, −1.31546706808683858316083502543,
0.52063861796521874514146860227, 2.33498589521772668832869168518, 3.42080614798821189122821568671, 4.35522333492228318089743862721, 5.06918255405823016559092409441, 6.44146515116958466555124712215, 6.94557114374622143199046209021, 8.146554917678966614730240716209, 8.628394099503993918349924193823, 9.336993336628447944925158702025