| L(s) = 1 | + 1.41i·2-s − 2.00·4-s − 2i·5-s − 2.82i·8-s + 2.82·10-s − 2.82i·11-s − 12.7·13-s + 4.00·16-s + 22i·17-s + 5.65·19-s + 4.00i·20-s + 4.00·22-s + 2.82i·23-s + 21·25-s − 18i·26-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.500·4-s − 0.400i·5-s − 0.353i·8-s + 0.282·10-s − 0.257i·11-s − 0.979·13-s + 0.250·16-s + 1.29i·17-s + 0.297·19-s + 0.200i·20-s + 0.181·22-s + 0.122i·23-s + 0.839·25-s − 0.692i·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.816 - 0.577i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.816 - 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.642741184\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.642741184\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 1.41iT \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 2iT - 25T^{2} \) |
| 11 | \( 1 + 2.82iT - 121T^{2} \) |
| 13 | \( 1 + 12.7T + 169T^{2} \) |
| 17 | \( 1 - 22iT - 289T^{2} \) |
| 19 | \( 1 - 5.65T + 361T^{2} \) |
| 23 | \( 1 - 2.82iT - 529T^{2} \) |
| 29 | \( 1 + 35.3iT - 841T^{2} \) |
| 31 | \( 1 - 33.9T + 961T^{2} \) |
| 37 | \( 1 - 64T + 1.36e3T^{2} \) |
| 41 | \( 1 + 20iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 44T + 1.84e3T^{2} \) |
| 47 | \( 1 + 68iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 18.3iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 100iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 52.3T + 3.72e3T^{2} \) |
| 67 | \( 1 - 120T + 4.48e3T^{2} \) |
| 71 | \( 1 - 8.48iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 74.9T + 5.32e3T^{2} \) |
| 79 | \( 1 - 92T + 6.24e3T^{2} \) |
| 83 | \( 1 + 112iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 20iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 26.8T + 9.40e3T^{2} \) |
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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
−9.872764245732024184339860828851, −9.078431810470269859053854790284, −8.211050586909745868008580443568, −7.58099468003946227607638784878, −6.51765581789409209756598516428, −5.74478191134501118672409423052, −4.77635180840864639505962931359, −3.92333577198126412740137709391, −2.47627188026566385276826774656, −0.805067473172015488345116749782,
0.847246425636990681161218393380, 2.43483088063024155219764390066, 3.11662978885447902695817070140, 4.51373838564450187056943290538, 5.15494788372360179783538372542, 6.49135433342551387115967289084, 7.34551252542406505638621817932, 8.198827491798137733819740041320, 9.494817162933210063268102336818, 9.656034277269365091970423982460
