| L(s) = 1 | + (0.777 − 1.18i)2-s + (0.780 + 1.54i)3-s + (−0.790 − 1.83i)4-s + (2.35 + 1.07i)5-s + (2.43 + 0.279i)6-s + (−0.243 + 0.828i)7-s + (−2.78 − 0.495i)8-s + (−1.78 + 2.41i)9-s + (3.10 − 1.94i)10-s + (1.57 − 1.81i)11-s + (2.22 − 2.65i)12-s + (3.50 − 1.02i)13-s + (0.789 + 0.931i)14-s + (0.176 + 4.47i)15-s + (−2.75 + 2.90i)16-s + (−6.33 + 0.910i)17-s + ⋯ |
| L(s) = 1 | + (0.549 − 0.835i)2-s + (0.450 + 0.892i)3-s + (−0.395 − 0.918i)4-s + (1.05 + 0.480i)5-s + (0.993 + 0.114i)6-s + (−0.0919 + 0.312i)7-s + (−0.984 − 0.175i)8-s + (−0.593 + 0.804i)9-s + (0.980 − 0.614i)10-s + (0.474 − 0.547i)11-s + (0.641 − 0.766i)12-s + (0.972 − 0.285i)13-s + (0.210 + 0.248i)14-s + (0.0454 + 1.15i)15-s + (−0.687 + 0.726i)16-s + (−1.53 + 0.220i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.960 + 0.279i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.960 + 0.279i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.02230 - 0.287886i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.02230 - 0.287886i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.777 + 1.18i)T \) |
| 3 | \( 1 + (-0.780 - 1.54i)T \) |
| 23 | \( 1 + (2.48 + 4.10i)T \) |
| good | 5 | \( 1 + (-2.35 - 1.07i)T + (3.27 + 3.77i)T^{2} \) |
| 7 | \( 1 + (0.243 - 0.828i)T + (-5.88 - 3.78i)T^{2} \) |
| 11 | \( 1 + (-1.57 + 1.81i)T + (-1.56 - 10.8i)T^{2} \) |
| 13 | \( 1 + (-3.50 + 1.02i)T + (10.9 - 7.02i)T^{2} \) |
| 17 | \( 1 + (6.33 - 0.910i)T + (16.3 - 4.78i)T^{2} \) |
| 19 | \( 1 + (-3.49 - 0.502i)T + (18.2 + 5.35i)T^{2} \) |
| 29 | \( 1 + (-1.56 + 0.225i)T + (27.8 - 8.17i)T^{2} \) |
| 31 | \( 1 + (2.56 + 3.99i)T + (-12.8 + 28.1i)T^{2} \) |
| 37 | \( 1 + (-0.589 - 1.29i)T + (-24.2 + 27.9i)T^{2} \) |
| 41 | \( 1 + (4.76 + 2.17i)T + (26.8 + 30.9i)T^{2} \) |
| 43 | \( 1 + (-2.54 + 3.96i)T + (-17.8 - 39.1i)T^{2} \) |
| 47 | \( 1 + 12.4T + 47T^{2} \) |
| 53 | \( 1 + (0.947 - 3.22i)T + (-44.5 - 28.6i)T^{2} \) |
| 59 | \( 1 + (6.35 - 1.86i)T + (49.6 - 31.8i)T^{2} \) |
| 61 | \( 1 + (10.2 - 6.60i)T + (25.3 - 55.4i)T^{2} \) |
| 67 | \( 1 + (-5.03 + 4.36i)T + (9.53 - 66.3i)T^{2} \) |
| 71 | \( 1 + (-9.79 - 11.3i)T + (-10.1 + 70.2i)T^{2} \) |
| 73 | \( 1 + (0.679 - 4.72i)T + (-70.0 - 20.5i)T^{2} \) |
| 79 | \( 1 + (-2.43 - 8.30i)T + (-66.4 + 42.7i)T^{2} \) |
| 83 | \( 1 + (2.03 + 4.45i)T + (-54.3 + 62.7i)T^{2} \) |
| 89 | \( 1 + (-7.28 + 11.3i)T + (-36.9 - 80.9i)T^{2} \) |
| 97 | \( 1 + (-7.74 + 16.9i)T + (-63.5 - 73.3i)T^{2} \) |
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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
−11.56937245414638446106382545569, −10.83722898508363602185245747324, −10.11444809489923339100039985680, −9.230529883754293970894072902796, −8.527353116173933180171867989900, −6.37008452847006798324673530572, −5.67884280800356220309573325177, −4.35717991928877480094325345401, −3.21972925994513262645142600939, −2.10616236773333502398840985987,
1.80667199254525211666930505544, 3.50726738486584248987872193924, 4.95141844674754053784900123180, 6.20127510917770200463101370890, 6.77582535285531718624947756181, 7.894686452823237734801056076861, 8.988883738594583203337142698581, 9.490196095269115672872480130682, 11.32059078135209976044081907837, 12.29893710920011707501676221079
