| L(s) = 1 | − 5.65i·3-s − 7·5-s − 11·7-s − 23.0·9-s + 3·11-s − 11.3i·13-s + 39.5i·15-s − 17·17-s − 19·19-s + 62.2i·21-s − 2·23-s + 24·25-s + 79.1i·27-s − 39.5i·29-s + 5.65i·31-s + ⋯ |
| L(s) = 1 | − 1.88i·3-s − 1.40·5-s − 1.57·7-s − 2.55·9-s + 0.272·11-s − 0.870i·13-s + 2.63i·15-s − 17-s − 19-s + 2.96i·21-s − 0.0869·23-s + 0.959·25-s + 2.93i·27-s − 1.36i·29-s + 0.182i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.06792195590\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.06792195590\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + 19T \) |
| good | 3 | \( 1 + 5.65iT - 9T^{2} \) |
| 5 | \( 1 + 7T + 25T^{2} \) |
| 7 | \( 1 + 11T + 49T^{2} \) |
| 11 | \( 1 - 3T + 121T^{2} \) |
| 13 | \( 1 + 11.3iT - 169T^{2} \) |
| 17 | \( 1 + 17T + 289T^{2} \) |
| 23 | \( 1 + 2T + 529T^{2} \) |
| 29 | \( 1 + 39.5iT - 841T^{2} \) |
| 31 | \( 1 - 5.65iT - 961T^{2} \) |
| 37 | \( 1 + 39.5iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 39.5iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 21T + 1.84e3T^{2} \) |
| 47 | \( 1 - 5T + 2.20e3T^{2} \) |
| 53 | \( 1 - 5.65iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 33.9iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 23T + 3.72e3T^{2} \) |
| 67 | \( 1 + 39.5iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 90.5iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 39T + 5.32e3T^{2} \) |
| 79 | \( 1 + 96.1iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 6T + 6.88e3T^{2} \) |
| 89 | \( 1 + 118. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 169. iT - 9.40e3T^{2} \) |
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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
−8.431825644146934926656598884659, −7.80743403561272948359982110156, −7.02308606120827680893592290552, −6.50506623033683166689465842016, −5.74116235583849464221598370865, −4.08562084291334047450648913169, −3.15777091138946962357321195091, −2.21794971763085243798823570837, −0.48773837777327157574429027387, −0.04288121501771759171089913368,
2.82158798408060234573502600703, 3.64239698037817293357079682601, 4.16865018399797391887960700234, 4.89053807554149028884187376236, 6.24383336522533258810506020173, 6.87427926301022186856762226721, 8.277787125946209191810643769546, 8.938076511698764801202415780583, 9.499504841946765614837880419047, 10.31150705842521774838538054939
