| L(s) = 1 | + (−1.45 + 1.37i)2-s + (2.66 + 1.38i)3-s + (0.234 − 3.99i)4-s + (8.07 − 4.66i)5-s + (−5.76 + 1.64i)6-s + (−4.91 − 2.83i)7-s + (5.13 + 6.13i)8-s + (5.18 + 7.35i)9-s + (−5.35 + 17.8i)10-s + (−1.85 + 3.21i)11-s + (6.13 − 10.3i)12-s + (−11.0 + 6.40i)13-s + (11.0 − 2.61i)14-s + (27.9 − 1.27i)15-s + (−15.8 − 1.87i)16-s + 11.1·17-s + ⋯ |
| L(s) = 1 | + (−0.727 + 0.686i)2-s + (0.887 + 0.460i)3-s + (0.0585 − 0.998i)4-s + (1.61 − 0.932i)5-s + (−0.961 + 0.274i)6-s + (−0.701 − 0.405i)7-s + (0.642 + 0.766i)8-s + (0.576 + 0.817i)9-s + (−0.535 + 1.78i)10-s + (−0.168 + 0.292i)11-s + (0.511 − 0.859i)12-s + (−0.853 + 0.492i)13-s + (0.788 − 0.186i)14-s + (1.86 − 0.0848i)15-s + (−0.993 − 0.116i)16-s + 0.657·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.817 - 0.576i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.817 - 0.576i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.23824 + 0.392485i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.23824 + 0.392485i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.45 - 1.37i)T \) |
| 3 | \( 1 + (-2.66 - 1.38i)T \) |
| good | 5 | \( 1 + (-8.07 + 4.66i)T + (12.5 - 21.6i)T^{2} \) |
| 7 | \( 1 + (4.91 + 2.83i)T + (24.5 + 42.4i)T^{2} \) |
| 11 | \( 1 + (1.85 - 3.21i)T + (-60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + (11.0 - 6.40i)T + (84.5 - 146. i)T^{2} \) |
| 17 | \( 1 - 11.1T + 289T^{2} \) |
| 19 | \( 1 + 13.1T + 361T^{2} \) |
| 23 | \( 1 + (20.2 - 11.6i)T + (264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + (14.6 + 8.47i)T + (420.5 + 728. i)T^{2} \) |
| 31 | \( 1 + (-3.32 + 1.91i)T + (480.5 - 832. i)T^{2} \) |
| 37 | \( 1 + 13.8iT - 1.36e3T^{2} \) |
| 41 | \( 1 + (-3.05 - 5.29i)T + (-840.5 + 1.45e3i)T^{2} \) |
| 43 | \( 1 + (11.3 - 19.7i)T + (-924.5 - 1.60e3i)T^{2} \) |
| 47 | \( 1 + (-49.8 - 28.8i)T + (1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 - 60.0iT - 2.80e3T^{2} \) |
| 59 | \( 1 + (55.3 + 95.8i)T + (-1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (73.2 + 42.3i)T + (1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-16.0 - 27.8i)T + (-2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 + 38.7iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 13.6T + 5.32e3T^{2} \) |
| 79 | \( 1 + (-4.14 - 2.39i)T + (3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 + (2.70 - 4.68i)T + (-3.44e3 - 5.96e3i)T^{2} \) |
| 89 | \( 1 - 98.2T + 7.92e3T^{2} \) |
| 97 | \( 1 + (-42.1 + 73.0i)T + (-4.70e3 - 8.14e3i)T^{2} \) |
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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
−14.41914586986407555538419443201, −13.76095651228618396032592497781, −12.73875281014212858963634474976, −10.33042720867251721103336819673, −9.678055075656515769065885158166, −9.072277636061519562106585388043, −7.67772499461887875177646183008, −6.13078993238453133958200198127, −4.77184709844073372165570918254, −1.99988829191963014191428235169,
2.15529094584554057710955814266, 3.10758309905602927907895775413, 6.15869812958417135390831407466, 7.36687520425697213561565896818, 8.848749825677590818796059349909, 9.838242645757673750685217740525, 10.42599883136019347052978614395, 12.27746587970631374919896529640, 13.15210331134535120566292238443, 14.04745700588373226096925635536
