L(s) = 1 | + (4.61 + 7.99i)5-s + (−5.33 − 3.07i)7-s + (−3.70 − 2.13i)11-s + (−0.869 − 1.50i)13-s − 12.3·17-s − 33.9i·19-s + (3.35 − 1.93i)23-s + (−30.1 + 52.1i)25-s + (17.8 − 30.9i)29-s + (38.8 − 22.4i)31-s − 56.8i·35-s + 32.7·37-s + (−21.8 − 37.8i)41-s + (−33.9 − 19.5i)43-s + (−39.8 − 23.0i)47-s + ⋯ |
L(s) = 1 | + (0.923 + 1.59i)5-s + (−0.761 − 0.439i)7-s + (−0.336 − 0.194i)11-s + (−0.0668 − 0.115i)13-s − 0.726·17-s − 1.78i·19-s + (0.145 − 0.0841i)23-s + (−1.20 + 2.08i)25-s + (0.615 − 1.06i)29-s + (1.25 − 0.723i)31-s − 1.62i·35-s + 0.884·37-s + (−0.533 − 0.923i)41-s + (−0.789 − 0.455i)43-s + (−0.848 − 0.489i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.659 + 0.751i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.659 + 0.751i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.578101761\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.578101761\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (-4.61 - 7.99i)T + (-12.5 + 21.6i)T^{2} \) |
| 7 | \( 1 + (5.33 + 3.07i)T + (24.5 + 42.4i)T^{2} \) |
| 11 | \( 1 + (3.70 + 2.13i)T + (60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + (0.869 + 1.50i)T + (-84.5 + 146. i)T^{2} \) |
| 17 | \( 1 + 12.3T + 289T^{2} \) |
| 19 | \( 1 + 33.9iT - 361T^{2} \) |
| 23 | \( 1 + (-3.35 + 1.93i)T + (264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + (-17.8 + 30.9i)T + (-420.5 - 728. i)T^{2} \) |
| 31 | \( 1 + (-38.8 + 22.4i)T + (480.5 - 832. i)T^{2} \) |
| 37 | \( 1 - 32.7T + 1.36e3T^{2} \) |
| 41 | \( 1 + (21.8 + 37.8i)T + (-840.5 + 1.45e3i)T^{2} \) |
| 43 | \( 1 + (33.9 + 19.5i)T + (924.5 + 1.60e3i)T^{2} \) |
| 47 | \( 1 + (39.8 + 23.0i)T + (1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 - 46.3T + 2.80e3T^{2} \) |
| 59 | \( 1 + (-23.2 + 13.4i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (23.4 - 40.6i)T + (-1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-56.9 + 32.9i)T + (2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 - 96.7iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 14.0T + 5.32e3T^{2} \) |
| 79 | \( 1 + (-34.3 - 19.8i)T + (3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 + (-81.7 - 47.1i)T + (3.44e3 + 5.96e3i)T^{2} \) |
| 89 | \( 1 - 81.8T + 7.92e3T^{2} \) |
| 97 | \( 1 + (7.99 - 13.8i)T + (-4.70e3 - 8.14e3i)T^{2} \) |
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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
−9.268780226879815432709473625171, −8.198968985960425623922615639251, −7.07079555425758960134573316434, −6.70429026860251624459970266728, −6.08181337603615028564024248871, −5.00255531250573155933592604796, −3.80107718121275238244938539449, −2.76423993544227592175591324495, −2.34165468615677444931789327780, −0.43924493086832258358779703722,
1.10483712779617980452660631171, 2.03156032408632842507639308337, 3.22554197126219033946388426611, 4.54355370511165820871871339140, 5.09836149939563299153911525240, 6.04614319312539860673724193901, 6.53051006868122389897753569582, 7.976191977983918387861164337369, 8.521237909901893139092360943079, 9.286379191861412659993427243061