| L(s) = 1 | + (−1.81 − 0.834i)2-s + (2.60 + 3.03i)4-s + 6.14·5-s + 0.590i·7-s + (−2.20 − 7.68i)8-s + (−11.1 − 5.12i)10-s − 17.4i·11-s + 1.78·13-s + (0.492 − 1.07i)14-s + (−2.39 + 15.8i)16-s + 16.9·17-s + 19.5i·19-s + (16.0 + 18.6i)20-s + (−14.5 + 31.7i)22-s − 7.93i·23-s + ⋯ |
| L(s) = 1 | + (−0.908 − 0.417i)2-s + (0.651 + 0.758i)4-s + 1.22·5-s + 0.0843i·7-s + (−0.276 − 0.961i)8-s + (−1.11 − 0.512i)10-s − 1.58i·11-s + 0.137·13-s + (0.0352 − 0.0766i)14-s + (−0.149 + 0.988i)16-s + 0.995·17-s + 1.02i·19-s + (0.801 + 0.932i)20-s + (−0.662 + 1.44i)22-s − 0.344i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.651 + 0.758i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.651 + 0.758i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.24674 - 0.572283i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.24674 - 0.572283i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.81 + 0.834i)T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 6.14T + 25T^{2} \) |
| 7 | \( 1 - 0.590iT - 49T^{2} \) |
| 11 | \( 1 + 17.4iT - 121T^{2} \) |
| 13 | \( 1 - 1.78T + 169T^{2} \) |
| 17 | \( 1 - 16.9T + 289T^{2} \) |
| 19 | \( 1 - 19.5iT - 361T^{2} \) |
| 23 | \( 1 + 7.93iT - 529T^{2} \) |
| 29 | \( 1 - 6.35T + 841T^{2} \) |
| 31 | \( 1 + 31.9iT - 961T^{2} \) |
| 37 | \( 1 - 58.2T + 1.36e3T^{2} \) |
| 41 | \( 1 + 5.33T + 1.68e3T^{2} \) |
| 43 | \( 1 + 39.1iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 11.1iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 35.8T + 2.80e3T^{2} \) |
| 59 | \( 1 + 24.1iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 75.8T + 3.72e3T^{2} \) |
| 67 | \( 1 - 36.7iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 87.8iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 60.0T + 5.32e3T^{2} \) |
| 79 | \( 1 - 37.1iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 76.2iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 27.5T + 7.92e3T^{2} \) |
| 97 | \( 1 + 26.1T + 9.40e3T^{2} \) |
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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
−11.06223949662974203475417989473, −10.20797742312569762005720964037, −9.530701105071051261562904511026, −8.575052880578264637556610041354, −7.75658739301432898918104620419, −6.27015114409732908387122046218, −5.70159207892842852460559138729, −3.64755381188907083687347467040, −2.42754725481253545003928390819, −1.01011066162593258630470300710,
1.38637031908208169064055739142, 2.56786143590704562724465100370, 4.80732327723897033318956040881, 5.81194413679062461907732411555, 6.81973587457409041642149168199, 7.63320093022708902014224334034, 8.896167561837056701433933816988, 9.777863113760489411685364574787, 10.10768684420993209120150188516, 11.26721686676266385585461633795
