| L(s) = 1 | − 3.74i·3-s + 9.79·7-s − 5·9-s + 4·11-s − 11.2i·13-s + 19.5·17-s + (5 + 18.3i)19-s − 36.6i·21-s + 9.79·23-s − 14.9i·27-s + 36.6i·29-s − 36.6i·31-s − 14.9i·33-s + 33.6i·37-s − 42·39-s + ⋯ |
| L(s) = 1 | − 1.24i·3-s + 1.39·7-s − 0.555·9-s + 0.363·11-s − 0.863i·13-s + 1.15·17-s + (0.263 + 0.964i)19-s − 1.74i·21-s + 0.425·23-s − 0.554i·27-s + 1.26i·29-s − 1.18i·31-s − 0.453i·33-s + 0.910i·37-s − 1.07·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.263 + 0.964i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.263 + 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.869750207\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.869750207\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 + (-5 - 18.3i)T \) |
| good | 3 | \( 1 + 3.74iT - 9T^{2} \) |
| 7 | \( 1 - 9.79T + 49T^{2} \) |
| 11 | \( 1 - 4T + 121T^{2} \) |
| 13 | \( 1 + 11.2iT - 169T^{2} \) |
| 17 | \( 1 - 19.5T + 289T^{2} \) |
| 23 | \( 1 - 9.79T + 529T^{2} \) |
| 29 | \( 1 - 36.6iT - 841T^{2} \) |
| 31 | \( 1 + 36.6iT - 961T^{2} \) |
| 37 | \( 1 - 33.6iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 36.6iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 68.5T + 1.84e3T^{2} \) |
| 47 | \( 1 + 9.79T + 2.20e3T^{2} \) |
| 53 | \( 1 - 56.1iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 73.3iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 100T + 3.72e3T^{2} \) |
| 67 | \( 1 - 11.2iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 36.6iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 19.5T + 5.32e3T^{2} \) |
| 79 | \( 1 - 109. iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 29.3T + 6.88e3T^{2} \) |
| 89 | \( 1 + 146. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 123. iT - 9.40e3T^{2} \) |
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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
−8.525767206827111756678519055962, −7.77877980677459189871117199390, −7.66311071986616680098218390781, −6.56600321867035081554993071161, −5.69760065346460010172508411576, −4.99848446232432113818303485637, −3.82116638962625667236239530692, −2.64522076764186682551414233619, −1.49646195132628934564199736437, −0.987822757220466272305843052073,
1.07255887754472467041908348877, 2.29864700018671812793133930127, 3.60743680221930865271393667519, 4.33048856393496567591629527446, 4.99651069815972846043679776409, 5.66286948546502727403214849648, 6.95229497945563282290846931269, 7.67098063787680329878106475814, 8.666789587816668900176633957463, 9.174228308381324093614826311975
