| L(s) = 1 | − 5.21·5-s − 2·7-s + 4.78·11-s − 1.38i·13-s + 14.6i·17-s + 26.7i·19-s − 18.0i·23-s + 2.23·25-s + 25.0·29-s + 39.5·31-s + 10.4·35-s − 26i·37-s − 28.8i·41-s + 9.52i·43-s − 80.2i·47-s + ⋯ |
| L(s) = 1 | − 1.04·5-s − 0.285·7-s + 0.434·11-s − 0.106i·13-s + 0.863i·17-s + 1.40i·19-s − 0.784i·23-s + 0.0895·25-s + 0.862·29-s + 1.27·31-s + 0.298·35-s − 0.702i·37-s − 0.702i·41-s + 0.221i·43-s − 1.70i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.985 + 0.169i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.985 + 0.169i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.02889120895\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.02889120895\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 5.21T + 25T^{2} \) |
| 7 | \( 1 + 2T + 49T^{2} \) |
| 11 | \( 1 - 4.78T + 121T^{2} \) |
| 13 | \( 1 + 1.38iT - 169T^{2} \) |
| 17 | \( 1 - 14.6iT - 289T^{2} \) |
| 19 | \( 1 - 26.7iT - 361T^{2} \) |
| 23 | \( 1 + 18.0iT - 529T^{2} \) |
| 29 | \( 1 - 25.0T + 841T^{2} \) |
| 31 | \( 1 - 39.5T + 961T^{2} \) |
| 37 | \( 1 + 26iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 28.8iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 9.52iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 80.2iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 9.79T + 2.80e3T^{2} \) |
| 59 | \( 1 + 73.5T + 3.48e3T^{2} \) |
| 61 | \( 1 - 67.5iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 102. iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 21.9iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 140.T + 5.32e3T^{2} \) |
| 79 | \( 1 - 0.476T + 6.24e3T^{2} \) |
| 83 | \( 1 - 31.3T + 6.88e3T^{2} \) |
| 89 | \( 1 - 13.1iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 69.2T + 9.40e3T^{2} \) |
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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
−9.120905441874276256536913052480, −8.201561968642241782458307260121, −7.981766181785851044714822643403, −6.84637786079404523200194505795, −6.26297469067651049005834959432, −5.28031081223411707514993957519, −4.10660921970596782502710887070, −3.78666280126439226405378129318, −2.62245126735371581964928891605, −1.29110372055988459744653970011,
0.008220077852859006086815901028, 1.13743897499460131022825237125, 2.70767931911331298430092427524, 3.40087994732330321390355568812, 4.48272100694802597222299165858, 4.96169089019443088368094985924, 6.33264880858294727790365230233, 6.81855677520326587727003118191, 7.77078564657295180773833221390, 8.239708775975371890855510255689
