| L(s) = 1 | − 1.73·3-s + 7.47i·5-s − 2.64i·7-s + 2.99·9-s + 13.5·11-s + 3.54i·13-s − 12.9i·15-s + 7.84·17-s + 11.1·19-s + 4.58i·21-s − 22.3i·23-s − 30.9·25-s − 5.19·27-s + 52.0i·29-s + 15.6i·31-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.49i·5-s − 0.377i·7-s + 0.333·9-s + 1.22·11-s + 0.272i·13-s − 0.863i·15-s + 0.461·17-s + 0.585·19-s + 0.218i·21-s − 0.973i·23-s − 1.23·25-s − 0.192·27-s + 1.79i·29-s + 0.505i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.117 - 0.993i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.117 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.440169482\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.440169482\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 1.73T \) |
| 7 | \( 1 + 2.64iT \) |
| good | 5 | \( 1 - 7.47iT - 25T^{2} \) |
| 11 | \( 1 - 13.5T + 121T^{2} \) |
| 13 | \( 1 - 3.54iT - 169T^{2} \) |
| 17 | \( 1 - 7.84T + 289T^{2} \) |
| 19 | \( 1 - 11.1T + 361T^{2} \) |
| 23 | \( 1 + 22.3iT - 529T^{2} \) |
| 29 | \( 1 - 52.0iT - 841T^{2} \) |
| 31 | \( 1 - 15.6iT - 961T^{2} \) |
| 37 | \( 1 - 22.1iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 36.1T + 1.68e3T^{2} \) |
| 43 | \( 1 + 81.8T + 1.84e3T^{2} \) |
| 47 | \( 1 - 66.3iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 83.6iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 47.7T + 3.48e3T^{2} \) |
| 61 | \( 1 - 102. iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 10.1T + 4.48e3T^{2} \) |
| 71 | \( 1 - 71.0iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 42.8T + 5.32e3T^{2} \) |
| 79 | \( 1 - 10.3iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 103.T + 6.88e3T^{2} \) |
| 89 | \( 1 + 60.9T + 7.92e3T^{2} \) |
| 97 | \( 1 - 52.2T + 9.40e3T^{2} \) |
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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
−10.56827996681720341594299120902, −9.947743048318251212495575365970, −8.918726656791534963346983587534, −7.63929957060963372127176304335, −6.70699894648263542795758196116, −6.48104049697018076095152036788, −5.11469434761439487342131339736, −3.88106535624261044908524142504, −2.97930105736918628782855397577, −1.35317613461243004293190931646,
0.63226628610679147646472297429, 1.72655615558483903178642465796, 3.65766055865999272147737970606, 4.65140127378155919663495228439, 5.51502188170142863891051149608, 6.23885710783675664003873261170, 7.54522712681671781086857511154, 8.393804298919124283149010294006, 9.388164393794344063494443868333, 9.731707051514897493422697114017
