| 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
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.4470895739\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4470895739\) |
| \(L(3)\) |
|
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
−10.97962159588147097139061176581, −10.23536139450935246434569148596, −8.302918170253086314552140972825, −8.104528871911236134630492685676, −7.63010264034747106622935490636, −6.20249309461562686878591516958, −5.04777850826941420432333828960, −3.68890912417119097328602323510, −2.66718703119923065581349218555, −1.02888819555824534285800994232,
0.14272126131083889913452826852, 2.31973833912136690193194768659, 3.64851983149401973492872741171, 4.51948227567807503114102543737, 5.19985958850412922096076595441, 7.10430480391362138649301682272, 7.78332577230208700304679599811, 8.742160592726702767474757558586, 9.453009859167576410550801797758, 10.76483300978353835818409279021
