| L(s) = 1 | − 3.65i·2-s + 4.18i·3-s + 2.64·4-s + 45.6i·5-s + 15.3·6-s + 34.3i·7-s − 68.1i·8-s + 63.4·9-s + 166.·10-s − 103.·11-s + 11.0i·12-s + 134.·13-s + 125.·14-s − 191.·15-s − 206.·16-s + 240.·17-s + ⋯ |
| L(s) = 1 | − 0.913i·2-s + 0.465i·3-s + 0.165·4-s + 1.82i·5-s + 0.425·6-s + 0.700i·7-s − 1.06i·8-s + 0.783·9-s + 1.66·10-s − 0.852·11-s + 0.0768i·12-s + 0.798·13-s + 0.640·14-s − 0.850·15-s − 0.807·16-s + 0.831·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.942 - 0.335i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.942 - 0.335i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.64771 + 0.284405i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.64771 + 0.284405i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (-1.74e3 + 619. i)T \) |
| good | 2 | \( 1 + 3.65iT - 16T^{2} \) |
| 3 | \( 1 - 4.18iT - 81T^{2} \) |
| 5 | \( 1 - 45.6iT - 625T^{2} \) |
| 7 | \( 1 - 34.3iT - 2.40e3T^{2} \) |
| 11 | \( 1 + 103.T + 1.46e4T^{2} \) |
| 13 | \( 1 - 134.T + 2.85e4T^{2} \) |
| 17 | \( 1 - 240.T + 8.35e4T^{2} \) |
| 19 | \( 1 + 100. iT - 1.30e5T^{2} \) |
| 23 | \( 1 + 475.T + 2.79e5T^{2} \) |
| 29 | \( 1 + 159. iT - 7.07e5T^{2} \) |
| 31 | \( 1 - 1.11e3T + 9.23e5T^{2} \) |
| 37 | \( 1 + 2.45e3iT - 1.87e6T^{2} \) |
| 41 | \( 1 - 1.06e3T + 2.82e6T^{2} \) |
| 47 | \( 1 + 2.93e3T + 4.87e6T^{2} \) |
| 53 | \( 1 + 825.T + 7.89e6T^{2} \) |
| 59 | \( 1 + 1.24e3T + 1.21e7T^{2} \) |
| 61 | \( 1 + 6.20e3iT - 1.38e7T^{2} \) |
| 67 | \( 1 - 265.T + 2.01e7T^{2} \) |
| 71 | \( 1 + 3.86e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 - 8.27e3iT - 2.83e7T^{2} \) |
| 79 | \( 1 - 4.95e3T + 3.89e7T^{2} \) |
| 83 | \( 1 + 1.35e4T + 4.74e7T^{2} \) |
| 89 | \( 1 - 5.35e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 - 3.19e3T + 8.85e7T^{2} \) |
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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
−15.48321337810202890363361235680, −14.14659854152982105033833912709, −12.65948796811111089698574410887, −11.35454833019285171933847209798, −10.54943014213267766268435913040, −9.788239382902760664303672127467, −7.52763847908721756140282321257, −6.16465808234016729311250674115, −3.63133345149936214909234479107, −2.39272410498991029173531096994,
1.26527506817634137306877896819, 4.60770522893687788492882239185, 5.99834149645641786465619607785, 7.64403608145323177224283317552, 8.360574355937320611138103104276, 10.09617295398892639322842721360, 11.89720712279267285681024875287, 12.98884801233211081220826619687, 13.83556209742295413741226771064, 15.61740967388467179459445621548
