| L(s) = 1 | − 2.82i·2-s − 8.00·4-s − 27.6i·5-s + 22.6i·8-s − 78.1·10-s − 68.7i·11-s − 234.·13-s + 64.0·16-s − 232. i·17-s − 358.·19-s + 220. i·20-s − 194.·22-s + 673. i·23-s − 137.·25-s + 661. i·26-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.500·4-s − 1.10i·5-s + 0.353i·8-s − 0.781·10-s − 0.568i·11-s − 1.38·13-s + 0.250·16-s − 0.803i·17-s − 0.992·19-s + 0.552i·20-s − 0.401·22-s + 1.27i·23-s − 0.220·25-s + 0.979i·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.816 - 0.577i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.816 - 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.3672691074\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3672691074\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 2.82iT \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 27.6iT - 625T^{2} \) |
| 11 | \( 1 + 68.7iT - 1.46e4T^{2} \) |
| 13 | \( 1 + 234.T + 2.85e4T^{2} \) |
| 17 | \( 1 + 232. iT - 8.35e4T^{2} \) |
| 19 | \( 1 + 358.T + 1.30e5T^{2} \) |
| 23 | \( 1 - 673. iT - 2.79e5T^{2} \) |
| 29 | \( 1 + 791. iT - 7.07e5T^{2} \) |
| 31 | \( 1 - 705.T + 9.23e5T^{2} \) |
| 37 | \( 1 + 1.37e3T + 1.87e6T^{2} \) |
| 41 | \( 1 - 56.5iT - 2.82e6T^{2} \) |
| 43 | \( 1 - 263.T + 3.41e6T^{2} \) |
| 47 | \( 1 - 2.28e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 + 2.32e3iT - 7.89e6T^{2} \) |
| 59 | \( 1 - 2.05e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 + 5.58e3T + 1.38e7T^{2} \) |
| 67 | \( 1 + 5.83e3T + 2.01e7T^{2} \) |
| 71 | \( 1 - 8.12e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 - 2.19e3T + 2.83e7T^{2} \) |
| 79 | \( 1 - 1.05e4T + 3.89e7T^{2} \) |
| 83 | \( 1 + 7.94e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 + 6.28e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 + 852.T + 8.85e7T^{2} \) |
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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
−9.601658141535603162139979786051, −8.999186076066552089711924263573, −8.162002447631974086077996941985, −7.28132669815846325253208264851, −5.94280519589832421077683676076, −4.98623878650201094408866861002, −4.40011448686517570763181365757, −3.12741784811143908481558938539, −2.04765591294413404722152372117, −0.859486867975118897449374699638,
0.098490862327920629275446187096, 2.00297621543008021574401088452, 3.02073605607790130468793861299, 4.26713487828885834689359960990, 5.08821400470585506257544885431, 6.35973686201132004560997047331, 6.81800206147210359448496505008, 7.61645816040894398598208032496, 8.506509661141081768002017018411, 9.454065314572263521640392136212
