| L(s) = 1 | + (−3 + 8.48i)3-s + 37.9·5-s + 37.9·7-s + (−62.9 − 50.9i)9-s + 134·11-s + 321. i·13-s + (−113. + 321. i)15-s + 237. i·17-s − 492. i·19-s + (−113. + 321. i)21-s − 643. i·23-s + 815·25-s + (620. − 381. i)27-s + 796.·29-s + 1.10e3·31-s + ⋯ |
| L(s) = 1 | + (−0.333 + 0.942i)3-s + 1.51·5-s + 0.774·7-s + (−0.777 − 0.628i)9-s + 1.10·11-s + 1.90i·13-s + (−0.505 + 1.43i)15-s + 0.822i·17-s − 1.36i·19-s + (−0.258 + 0.730i)21-s − 1.21i·23-s + 1.30·25-s + (0.851 − 0.523i)27-s + 0.947·29-s + 1.14·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.333 - 0.942i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.333 - 0.942i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(2.857100630\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.857100630\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (3 - 8.48i)T \) |
| good | 5 | \( 1 - 37.9T + 625T^{2} \) |
| 7 | \( 1 - 37.9T + 2.40e3T^{2} \) |
| 11 | \( 1 - 134T + 1.46e4T^{2} \) |
| 13 | \( 1 - 321. iT - 2.85e4T^{2} \) |
| 17 | \( 1 - 237. iT - 8.35e4T^{2} \) |
| 19 | \( 1 + 492. iT - 1.30e5T^{2} \) |
| 23 | \( 1 + 643. iT - 2.79e5T^{2} \) |
| 29 | \( 1 - 796.T + 7.07e5T^{2} \) |
| 31 | \( 1 - 1.10e3T + 9.23e5T^{2} \) |
| 37 | \( 1 + 1.60e3iT - 1.87e6T^{2} \) |
| 41 | \( 1 - 1.28e3iT - 2.82e6T^{2} \) |
| 43 | \( 1 - 424. iT - 3.41e6T^{2} \) |
| 47 | \( 1 - 2.57e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 + 1.63e3T + 7.89e6T^{2} \) |
| 59 | \( 1 + 1.38e3T + 1.21e7T^{2} \) |
| 61 | \( 1 - 321. iT - 1.38e7T^{2} \) |
| 67 | \( 1 + 186. iT - 2.01e7T^{2} \) |
| 71 | \( 1 - 1.93e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 - 4.89e3T + 2.83e7T^{2} \) |
| 79 | \( 1 + 1.47e3T + 3.89e7T^{2} \) |
| 83 | \( 1 + 5.91e3T + 4.74e7T^{2} \) |
| 89 | \( 1 - 9.06e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 - 5.85e3T + 8.85e7T^{2} \) |
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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
−10.90635547166415262161214258457, −9.858608689660043980160950201211, −9.208174991247123949258691537452, −8.587929114066330746201998701745, −6.58591718895229054336555355856, −6.24908175476181567170742756798, −4.83394479051005586261730082593, −4.27962885669750693861051280121, −2.48426995633550173275920915321, −1.30131357889534417322863241613,
0.985631929268675414358453245315, 1.76862549458342717227578103905, 3.05906629281221304184114089871, 5.07849028101878372972399635501, 5.75443877622999579663147582544, 6.55683486557240778684909366046, 7.74214755623368871971846469302, 8.529758501539795538801682974979, 9.782228751564445990189973180640, 10.44936758558474497545603450335
