L(s) = 1 | + 13.4i·7-s + 17.6i·11-s − 7.48i·13-s − 16.9·17-s + 10.9·19-s − 21.9·23-s − 47.3i·29-s − 16.9·31-s + 5.53i·37-s − 66.3i·41-s + 38.9i·43-s − 32.5·47-s − 132.·49-s + 11.2·53-s − 31.8i·59-s + ⋯ |
L(s) = 1 | + 1.92i·7-s + 1.60i·11-s − 0.575i·13-s − 0.998·17-s + 0.577·19-s − 0.952·23-s − 1.63i·29-s − 0.547·31-s + 0.149i·37-s − 1.61i·41-s + 0.906i·43-s − 0.692·47-s − 2.71·49-s + 0.212·53-s − 0.540i·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.988 - 0.151i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.988 - 0.151i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.8701464469\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8701464469\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - 13.4iT - 49T^{2} \) |
| 11 | \( 1 - 17.6iT - 121T^{2} \) |
| 13 | \( 1 + 7.48iT - 169T^{2} \) |
| 17 | \( 1 + 16.9T + 289T^{2} \) |
| 19 | \( 1 - 10.9T + 361T^{2} \) |
| 23 | \( 1 + 21.9T + 529T^{2} \) |
| 29 | \( 1 + 47.3iT - 841T^{2} \) |
| 31 | \( 1 + 16.9T + 961T^{2} \) |
| 37 | \( 1 - 5.53iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 66.3iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 38.9iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 32.5T + 2.20e3T^{2} \) |
| 53 | \( 1 - 11.2T + 2.80e3T^{2} \) |
| 59 | \( 1 + 31.8iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 46.9T + 3.72e3T^{2} \) |
| 67 | \( 1 - 76iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 77.7iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 94.9iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 6.92T + 6.24e3T^{2} \) |
| 83 | \( 1 + 62.1T + 6.88e3T^{2} \) |
| 89 | \( 1 + 62.2iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 124. iT - 9.40e3T^{2} \) |
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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
−10.02094721504013907658024470244, −9.546870883797987261070570109970, −8.667660243550979176693853795557, −7.900448364332534805702127514259, −6.85822946620740452407187844746, −5.87126306981441742552251554929, −5.17270857613587212435919700680, −4.12186670954146915772302108931, −2.59495869733104063950099638473, −1.98501816334522092710717941286,
0.27561749974600317486634808155, 1.47004919948285095665611445927, 3.24977788739380644352820547596, 3.96686182859344493301768107409, 4.95802010871997178466410763448, 6.22775637406974488053026851173, 6.93846238696681440619002711399, 7.77781113584013655484066583484, 8.637967309967409253990878668323, 9.554655568356524525218679010077