| L(s) = 1 | + 2.65i·3-s + 0.895·7-s + 1.96·9-s − 1.97i·11-s + 4.58i·13-s + 13.5i·17-s + 13.3·19-s + 2.37i·21-s − 8.61i·23-s + 29.0i·27-s − 15.3i·29-s + (5.19 − 30.5i)31-s + 5.23·33-s − 47.2i·37-s − 12.1·39-s + ⋯ |
| L(s) = 1 | + 0.883i·3-s + 0.127·7-s + 0.218·9-s − 0.179i·11-s + 0.352i·13-s + 0.796i·17-s + 0.704·19-s + 0.113i·21-s − 0.374i·23-s + 1.07i·27-s − 0.530i·29-s + (0.167 − 0.985i)31-s + 0.158·33-s − 1.27i·37-s − 0.311·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.167 - 0.985i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3100 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.167 - 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.326623311\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.326623311\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 31 | \( 1 + (-5.19 + 30.5i)T \) |
| good | 3 | \( 1 - 2.65iT - 9T^{2} \) |
| 7 | \( 1 - 0.895T + 49T^{2} \) |
| 11 | \( 1 + 1.97iT - 121T^{2} \) |
| 13 | \( 1 - 4.58iT - 169T^{2} \) |
| 17 | \( 1 - 13.5iT - 289T^{2} \) |
| 19 | \( 1 - 13.3T + 361T^{2} \) |
| 23 | \( 1 + 8.61iT - 529T^{2} \) |
| 29 | \( 1 + 15.3iT - 841T^{2} \) |
| 37 | \( 1 + 47.2iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 75.1T + 1.68e3T^{2} \) |
| 43 | \( 1 - 62.5iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 82.9T + 2.20e3T^{2} \) |
| 53 | \( 1 - 60.1iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 62.3T + 3.48e3T^{2} \) |
| 61 | \( 1 - 10.2iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 29.8T + 4.48e3T^{2} \) |
| 71 | \( 1 - 38.0T + 5.04e3T^{2} \) |
| 73 | \( 1 + 2.30iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 110. iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 26.9iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 55.6iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 13.5T + 9.40e3T^{2} \) |
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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
−8.952561736800682774056028971369, −7.85131047580986359391604939382, −7.39141334534485552254780457057, −6.21142977848522078352489030970, −5.69803246368368902426215273363, −4.53626555069398941604918049355, −4.20390004520333770298638534702, −3.24102828368363984011974338367, −2.15999754226588676084034282987, −0.929686347897694807002398143619,
0.65821417597100936394006908643, 1.51681074350022151883952764335, 2.54311232482411135059669849951, 3.45814705152294826429672430243, 4.57529518331708104998311251107, 5.32338033780589287102652721213, 6.20230349933521478593260179269, 7.07584540001482867672244502880, 7.43146537885486137022744048327, 8.206527312030231799191495858888
