| L(s) = 1 | − 2.23·5-s − 4.03i·7-s + 5.56i·11-s + 10.5·13-s + 18.5·17-s + 22.4i·19-s − 28.1i·23-s + 5.00·25-s − 29.7·29-s + 41.2i·31-s + 9.02i·35-s − 44.3·37-s − 15.2·41-s − 24.9i·43-s − 48.5i·47-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.576i·7-s + 0.505i·11-s + 0.810·13-s + 1.08·17-s + 1.18i·19-s − 1.22i·23-s + 0.200·25-s − 1.02·29-s + 1.33i·31-s + 0.257i·35-s − 1.19·37-s − 0.371·41-s − 0.581i·43-s − 1.03i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.866 - 0.5i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.866 - 0.5i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.788860571\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.788860571\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + 2.23T \) |
| good | 7 | \( 1 + 4.03iT - 49T^{2} \) |
| 11 | \( 1 - 5.56iT - 121T^{2} \) |
| 13 | \( 1 - 10.5T + 169T^{2} \) |
| 17 | \( 1 - 18.5T + 289T^{2} \) |
| 19 | \( 1 - 22.4iT - 361T^{2} \) |
| 23 | \( 1 + 28.1iT - 529T^{2} \) |
| 29 | \( 1 + 29.7T + 841T^{2} \) |
| 31 | \( 1 - 41.2iT - 961T^{2} \) |
| 37 | \( 1 + 44.3T + 1.36e3T^{2} \) |
| 41 | \( 1 + 15.2T + 1.68e3T^{2} \) |
| 43 | \( 1 + 24.9iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 48.5iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 40.4T + 2.80e3T^{2} \) |
| 59 | \( 1 - 24.4iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 53.2T + 3.72e3T^{2} \) |
| 67 | \( 1 - 18.3iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 92.9iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 109.T + 5.32e3T^{2} \) |
| 79 | \( 1 - 53.3iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 62.9iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 67.7T + 7.92e3T^{2} \) |
| 97 | \( 1 - 126.T + 9.40e3T^{2} \) |
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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
−8.748445297908890186236500627084, −8.250058213317496299675779877702, −7.36381471526307565974262641795, −6.78031231837733562583767481779, −5.76778952131925997668085044437, −4.95420373152039063416969901339, −3.85812769107501694256810343913, −3.43274533324429910823890862847, −1.94895532672763318912263713818, −0.866847378391972623539558882334,
0.59236641596258901636836481674, 1.85753861913643585112765789070, 3.14522829226164944752725213987, 3.71701458000530092657841374806, 4.91288475533797678978282926897, 5.68184616031730049053200808715, 6.37502760681153736824781436298, 7.47739876328674115547443991679, 7.962912841267206061850783312816, 8.941573614354623248766236496935
