| L(s) = 1 | + (−2.81 − 1.04i)3-s + (4.33 + 2.48i)5-s + 10.8i·7-s + (6.83 + 5.85i)9-s + 15.2·11-s − 1.83·13-s + (−9.62 − 11.5i)15-s + 20.4·17-s + 13.6i·19-s + (11.3 − 30.6i)21-s − 19.4·23-s + (12.6 + 21.5i)25-s + (−13.1 − 23.6i)27-s + 24.5·29-s − 50.8·31-s + ⋯ |
| L(s) = 1 | + (−0.937 − 0.347i)3-s + (0.867 + 0.496i)5-s + 1.55i·7-s + (0.758 + 0.651i)9-s + 1.39·11-s − 0.141·13-s + (−0.641 − 0.767i)15-s + 1.20·17-s + 0.719i·19-s + (0.539 − 1.45i)21-s − 0.845·23-s + (0.506 + 0.862i)25-s + (−0.485 − 0.874i)27-s + 0.847·29-s − 1.64·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0888 - 0.996i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.0888 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.709260102\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.709260102\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (2.81 + 1.04i)T \) |
| 5 | \( 1 + (-4.33 - 2.48i)T \) |
| good | 7 | \( 1 - 10.8iT - 49T^{2} \) |
| 11 | \( 1 - 15.2T + 121T^{2} \) |
| 13 | \( 1 + 1.83T + 169T^{2} \) |
| 17 | \( 1 - 20.4T + 289T^{2} \) |
| 19 | \( 1 - 13.6iT - 361T^{2} \) |
| 23 | \( 1 + 19.4T + 529T^{2} \) |
| 29 | \( 1 - 24.5T + 841T^{2} \) |
| 31 | \( 1 + 50.8T + 961T^{2} \) |
| 37 | \( 1 - 16.2T + 1.36e3T^{2} \) |
| 41 | \( 1 + 42.3iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 26.2T + 1.84e3T^{2} \) |
| 47 | \( 1 + 58.3T + 2.20e3T^{2} \) |
| 53 | \( 1 - 63.7iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 31.5T + 3.48e3T^{2} \) |
| 61 | \( 1 + 81.8iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 50.6T + 4.48e3T^{2} \) |
| 71 | \( 1 - 135. iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 33.7iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 81.8T + 6.24e3T^{2} \) |
| 83 | \( 1 + 75.6iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 124. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 58.1iT - 9.40e3T^{2} \) |
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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
−9.913055701329192969764816373964, −9.433267626887099518032735421912, −8.406374211791727734695310232230, −7.28609285548177488187929212069, −6.32147444663069656517184363986, −5.84298782590133719148602925520, −5.19341719335625020847914508176, −3.70656325589266616993875611814, −2.30176773370651484909398385685, −1.40725615175981556404449020906,
0.69003500776489016862850177939, 1.51739642429977616439314191333, 3.57681330989485362788277626207, 4.37118102325337293771471607686, 5.22381427770527659037894498129, 6.25946102453476730189242411422, 6.83116640047829534703009172274, 7.82140412102358129878754539367, 9.152457491063255944779705778221, 9.788560254525709306410596685320
