| L(s) = 1 | + 0.0436·3-s − 2.23i·5-s + 2.33i·7-s − 8.99·9-s − 16.3i·11-s + 1.02·13-s − 0.0976i·15-s + 17.1i·17-s + 23.2i·19-s + 0.101i·21-s + (−2.92 + 22.8i)23-s − 5.00·25-s − 0.786·27-s − 5.00·29-s + 26.6·31-s + ⋯ |
| L(s) = 1 | + 0.0145·3-s − 0.447i·5-s + 0.333i·7-s − 0.999·9-s − 1.49i·11-s + 0.0789·13-s − 0.00651i·15-s + 1.00i·17-s + 1.22i·19-s + 0.00485i·21-s + (−0.127 + 0.991i)23-s − 0.200·25-s − 0.0291·27-s − 0.172·29-s + 0.859·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 + 0.127i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.991 + 0.127i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.662835670\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.662835670\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + 2.23iT \) |
| 23 | \( 1 + (2.92 - 22.8i)T \) |
| good | 3 | \( 1 - 0.0436T + 9T^{2} \) |
| 7 | \( 1 - 2.33iT - 49T^{2} \) |
| 11 | \( 1 + 16.3iT - 121T^{2} \) |
| 13 | \( 1 - 1.02T + 169T^{2} \) |
| 17 | \( 1 - 17.1iT - 289T^{2} \) |
| 19 | \( 1 - 23.2iT - 361T^{2} \) |
| 29 | \( 1 + 5.00T + 841T^{2} \) |
| 31 | \( 1 - 26.6T + 961T^{2} \) |
| 37 | \( 1 - 2.49iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 69.6T + 1.68e3T^{2} \) |
| 43 | \( 1 + 26.6iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 40.1T + 2.20e3T^{2} \) |
| 53 | \( 1 + 53.7iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 3.94T + 3.48e3T^{2} \) |
| 61 | \( 1 + 44.9iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 93.6iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 97.9T + 5.04e3T^{2} \) |
| 73 | \( 1 - 38.8T + 5.32e3T^{2} \) |
| 79 | \( 1 - 103. iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 59.3iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 130. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 142. iT - 9.40e3T^{2} \) |
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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
−8.906862046827085273801666067889, −8.259010883469469681104467255105, −7.85098688257119467465598550744, −6.33396046478498130728428600922, −5.86435489500452793901600658791, −5.22309166687093598424214531436, −3.86668762465556401265964449668, −3.23379243960458871130343031436, −1.99110832278896769632426364029, −0.70162178706136146337683827066,
0.67464478834258968312360569084, 2.34129306402285698157288182218, 2.88477146226456992039879542154, 4.26249206176792532950457517787, 4.87738468267896693405601093523, 5.94548473662820962371280967026, 6.85307461732933103644189881492, 7.36466355944145728706447175142, 8.272674413756891277461459107612, 9.202176257566355771415946415668
