| L(s) = 1 | + (1.59 + 1.20i)2-s + (−0.262 − 2.98i)3-s + (1.08 + 3.84i)4-s + (1.10 + 1.90i)5-s + (3.18 − 5.08i)6-s + (−7.23 − 4.17i)7-s + (−2.90 + 7.45i)8-s + (−8.86 + 1.56i)9-s + (−0.544 + 4.36i)10-s + (4.54 + 2.62i)11-s + (11.2 − 4.26i)12-s + (−7.37 − 12.7i)13-s + (−6.50 − 15.3i)14-s + (5.40 − 3.79i)15-s + (−13.6 + 8.38i)16-s + 28.2·17-s + ⋯ |
| L(s) = 1 | + (0.797 + 0.603i)2-s + (−0.0874 − 0.996i)3-s + (0.272 + 0.962i)4-s + (0.220 + 0.381i)5-s + (0.531 − 0.847i)6-s + (−1.03 − 0.597i)7-s + (−0.363 + 0.931i)8-s + (−0.984 + 0.174i)9-s + (−0.0544 + 0.436i)10-s + (0.413 + 0.238i)11-s + (0.934 − 0.355i)12-s + (−0.567 − 0.982i)13-s + (−0.464 − 1.09i)14-s + (0.360 − 0.252i)15-s + (−0.851 + 0.524i)16-s + 1.66·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.962 - 0.272i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.962 - 0.272i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.37443 + 0.191141i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.37443 + 0.191141i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.59 - 1.20i)T \) |
| 3 | \( 1 + (0.262 + 2.98i)T \) |
| good | 5 | \( 1 + (-1.10 - 1.90i)T + (-12.5 + 21.6i)T^{2} \) |
| 7 | \( 1 + (7.23 + 4.17i)T + (24.5 + 42.4i)T^{2} \) |
| 11 | \( 1 + (-4.54 - 2.62i)T + (60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + (7.37 + 12.7i)T + (-84.5 + 146. i)T^{2} \) |
| 17 | \( 1 - 28.2T + 289T^{2} \) |
| 19 | \( 1 - 19.1iT - 361T^{2} \) |
| 23 | \( 1 + (3.16 - 1.82i)T + (264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + (12.3 - 21.3i)T + (-420.5 - 728. i)T^{2} \) |
| 31 | \( 1 + (-32.9 + 19.0i)T + (480.5 - 832. i)T^{2} \) |
| 37 | \( 1 + 4.21T + 1.36e3T^{2} \) |
| 41 | \( 1 + (9.92 + 17.1i)T + (-840.5 + 1.45e3i)T^{2} \) |
| 43 | \( 1 + (20.1 + 11.6i)T + (924.5 + 1.60e3i)T^{2} \) |
| 47 | \( 1 + (25.8 + 14.9i)T + (1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 + 32.1T + 2.80e3T^{2} \) |
| 59 | \( 1 + (-7.96 + 4.59i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (40.8 - 70.7i)T + (-1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (6.86 - 3.96i)T + (2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 - 62.9iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 33.3T + 5.32e3T^{2} \) |
| 79 | \( 1 + (53.7 + 31.0i)T + (3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 + (-103. - 59.4i)T + (3.44e3 + 5.96e3i)T^{2} \) |
| 89 | \( 1 + 107.T + 7.92e3T^{2} \) |
| 97 | \( 1 + (-1.78 + 3.09i)T + (-4.70e3 - 8.14e3i)T^{2} \) |
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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
−16.44302169975857022308413976217, −14.78887852780955697768880376892, −13.93082854410623222521925640399, −12.79379292216729107171010796500, −12.05831575962442006054155261474, −10.15288361823552662972634822463, −7.972325460367646764847661607569, −6.89513480434773642075771726054, −5.74082756332321029974359179136, −3.21588844703711270142071858410,
3.19808680226487605547216138440, 4.91077347151932882047960833283, 6.26974199994338523016077774416, 9.227685699329009163922477980040, 9.902875549960629951201439831502, 11.47980989478543144397924079448, 12.44450352535630592411105165915, 13.86048589350893193140317568438, 14.94331629740759141920900019137, 16.03324153292167928807970036358
