| L(s) = 1 | + (0.619 − 2.93i)3-s + (−4.48 + 2.21i)5-s − 4.63i·7-s + (−8.23 − 3.63i)9-s + 15.0·11-s + 3.35·13-s + (3.71 + 14.5i)15-s + 11.1·17-s − 22.7i·19-s + (−13.5 − 2.87i)21-s + 36.1·23-s + (15.2 − 19.8i)25-s + (−15.7 + 21.9i)27-s − 21.5·29-s − 28.3·31-s + ⋯ |
| L(s) = 1 | + (0.206 − 0.978i)3-s + (−0.896 + 0.442i)5-s − 0.661i·7-s + (−0.914 − 0.404i)9-s + 1.37·11-s + 0.258·13-s + (0.247 + 0.968i)15-s + 0.655·17-s − 1.19i·19-s + (−0.647 − 0.136i)21-s + 1.57·23-s + (0.608 − 0.793i)25-s + (−0.584 + 0.811i)27-s − 0.743·29-s − 0.915·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.871 + 0.490i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.871 + 0.490i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.283886981\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.283886981\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.619 + 2.93i)T \) |
| 5 | \( 1 + (4.48 - 2.21i)T \) |
| good | 7 | \( 1 + 4.63iT - 49T^{2} \) |
| 11 | \( 1 - 15.0T + 121T^{2} \) |
| 13 | \( 1 - 3.35T + 169T^{2} \) |
| 17 | \( 1 - 11.1T + 289T^{2} \) |
| 19 | \( 1 + 22.7iT - 361T^{2} \) |
| 23 | \( 1 - 36.1T + 529T^{2} \) |
| 29 | \( 1 + 21.5T + 841T^{2} \) |
| 31 | \( 1 + 28.3T + 961T^{2} \) |
| 37 | \( 1 + 69.5T + 1.36e3T^{2} \) |
| 41 | \( 1 + 62.5iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 55.2T + 1.84e3T^{2} \) |
| 47 | \( 1 + 36.5T + 2.20e3T^{2} \) |
| 53 | \( 1 - 29.2iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 8.07T + 3.48e3T^{2} \) |
| 61 | \( 1 - 10.5iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 91.8T + 4.48e3T^{2} \) |
| 71 | \( 1 + 79.2iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 59.0iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 86.3T + 6.24e3T^{2} \) |
| 83 | \( 1 - 10.0iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 17.8iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 139. iT - 9.40e3T^{2} \) |
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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
−9.117492236150830093794749355232, −8.717295478776562429318963949742, −7.45777031851942575886926188617, −7.12197086656815947664179039732, −6.43113686307388481074045266440, −5.09832148395111543724494538247, −3.77514765287760970802682742736, −3.16927699704077026667877220984, −1.58417157322828374216683686217, −0.42617155303761429496365478321,
1.47837462438100982721699932284, 3.30051942262440077801174644643, 3.76334280989771906287049260114, 4.87969809995765454519605197842, 5.61991031465743237164550810050, 6.79517709580994352232942654836, 7.925478923845928672660018827080, 8.730109152535003687562663737786, 9.172669794198534595239869217003, 10.06374851881346128440489010974
