| L(s) = 1 | + (−0.841 − 0.540i)3-s + (−0.380 + 0.832i)5-s + (2.32 + 0.682i)7-s + (0.415 + 0.909i)9-s + (−2.20 + 2.54i)11-s + (−3.87 + 1.13i)13-s + (0.770 − 0.494i)15-s + (0.875 + 6.09i)17-s + (0.929 − 6.46i)19-s + (−1.58 − 1.83i)21-s + (4.54 + 1.53i)23-s + (2.72 + 3.14i)25-s + (0.142 − 0.989i)27-s + (0.505 + 3.51i)29-s + (−3.45 + 2.22i)31-s + ⋯ |
| L(s) = 1 | + (−0.485 − 0.312i)3-s + (−0.170 + 0.372i)5-s + (0.878 + 0.257i)7-s + (0.138 + 0.303i)9-s + (−0.666 + 0.768i)11-s + (−1.07 + 0.315i)13-s + (0.198 − 0.127i)15-s + (0.212 + 1.47i)17-s + (0.213 − 1.48i)19-s + (−0.346 − 0.399i)21-s + (0.947 + 0.320i)23-s + (0.545 + 0.629i)25-s + (0.0273 − 0.190i)27-s + (0.0939 + 0.653i)29-s + (−0.620 + 0.398i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.309 - 0.950i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.309 - 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.863459 + 0.627031i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.863459 + 0.627031i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.841 + 0.540i)T \) |
| 23 | \( 1 + (-4.54 - 1.53i)T \) |
| good | 5 | \( 1 + (0.380 - 0.832i)T + (-3.27 - 3.77i)T^{2} \) |
| 7 | \( 1 + (-2.32 - 0.682i)T + (5.88 + 3.78i)T^{2} \) |
| 11 | \( 1 + (2.20 - 2.54i)T + (-1.56 - 10.8i)T^{2} \) |
| 13 | \( 1 + (3.87 - 1.13i)T + (10.9 - 7.02i)T^{2} \) |
| 17 | \( 1 + (-0.875 - 6.09i)T + (-16.3 + 4.78i)T^{2} \) |
| 19 | \( 1 + (-0.929 + 6.46i)T + (-18.2 - 5.35i)T^{2} \) |
| 29 | \( 1 + (-0.505 - 3.51i)T + (-27.8 + 8.17i)T^{2} \) |
| 31 | \( 1 + (3.45 - 2.22i)T + (12.8 - 28.1i)T^{2} \) |
| 37 | \( 1 + (-3.49 - 7.66i)T + (-24.2 + 27.9i)T^{2} \) |
| 41 | \( 1 + (4.08 - 8.94i)T + (-26.8 - 30.9i)T^{2} \) |
| 43 | \( 1 + (-5.12 - 3.29i)T + (17.8 + 39.1i)T^{2} \) |
| 47 | \( 1 - 3.53T + 47T^{2} \) |
| 53 | \( 1 + (9.76 + 2.86i)T + (44.5 + 28.6i)T^{2} \) |
| 59 | \( 1 + (-8.04 + 2.36i)T + (49.6 - 31.8i)T^{2} \) |
| 61 | \( 1 + (-7.43 + 4.77i)T + (25.3 - 55.4i)T^{2} \) |
| 67 | \( 1 + (2.85 + 3.29i)T + (-9.53 + 66.3i)T^{2} \) |
| 71 | \( 1 + (-2.05 - 2.36i)T + (-10.1 + 70.2i)T^{2} \) |
| 73 | \( 1 + (-2.20 + 15.3i)T + (-70.0 - 20.5i)T^{2} \) |
| 79 | \( 1 + (11.4 - 3.36i)T + (66.4 - 42.7i)T^{2} \) |
| 83 | \( 1 + (4.10 + 8.99i)T + (-54.3 + 62.7i)T^{2} \) |
| 89 | \( 1 + (-2.70 - 1.73i)T + (36.9 + 80.9i)T^{2} \) |
| 97 | \( 1 + (-3.96 + 8.68i)T + (-63.5 - 73.3i)T^{2} \) |
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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
−11.10044647493173429739437881965, −10.25439663938162327659526140649, −9.231525854901676169059378626418, −8.115642625255142533646165462420, −7.32279402628939327507301863433, −6.58526661890090360107821766467, −5.14810623152859350581687436335, −4.72218355568477400648209852212, −2.95019871821457810252130924982, −1.65701789401681714735851613590,
0.67530995777763123328279345537, 2.59927395200482392932140906773, 4.07703430601549344761884094220, 5.11547802461871151324430304354, 5.63778980169314810205271736779, 7.19067556562859051596540745028, 7.85580412797112380408494560446, 8.844317824906056727814889948382, 9.872343568298136596464329449899, 10.66161977873943192951807259461
