L(s) = 1 | + (−2.78 − 4.15i)5-s − 0.701i·7-s − 0.462i·11-s + 14.1i·13-s − 14.3·17-s + 5.37·19-s + 41.6·23-s + (−9.51 + 23.1i)25-s − 9.34i·29-s + 21.8·31-s + (−2.91 + 1.95i)35-s + 34.0i·37-s − 22.3i·41-s − 55.9i·43-s + 7.64·47-s + ⋯ |
L(s) = 1 | + (−0.556 − 0.830i)5-s − 0.100i·7-s − 0.0420i·11-s + 1.08i·13-s − 0.846·17-s + 0.283·19-s + 1.81·23-s + (−0.380 + 0.924i)25-s − 0.322i·29-s + 0.705·31-s + (−0.0833 + 0.0558i)35-s + 0.921i·37-s − 0.545i·41-s − 1.30i·43-s + 0.162·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.556 + 0.830i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.556 + 0.830i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.574639256\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.574639256\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (2.78 + 4.15i)T \) |
good | 7 | \( 1 + 0.701iT - 49T^{2} \) |
| 11 | \( 1 + 0.462iT - 121T^{2} \) |
| 13 | \( 1 - 14.1iT - 169T^{2} \) |
| 17 | \( 1 + 14.3T + 289T^{2} \) |
| 19 | \( 1 - 5.37T + 361T^{2} \) |
| 23 | \( 1 - 41.6T + 529T^{2} \) |
| 29 | \( 1 + 9.34iT - 841T^{2} \) |
| 31 | \( 1 - 21.8T + 961T^{2} \) |
| 37 | \( 1 - 34.0iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 22.3iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 55.9iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 7.64T + 2.20e3T^{2} \) |
| 53 | \( 1 - 36.2T + 2.80e3T^{2} \) |
| 59 | \( 1 + 35.0iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 58.8T + 3.72e3T^{2} \) |
| 67 | \( 1 - 61.8iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 43.6iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 50.7iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 96.1T + 6.24e3T^{2} \) |
| 83 | \( 1 + 26.0T + 6.88e3T^{2} \) |
| 89 | \( 1 + 125. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 117. iT - 9.40e3T^{2} \) |
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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
−8.842603713035218766022530648174, −8.604738917665492329492466162805, −7.36692428191597483700356475202, −6.87907369786959283392536130194, −5.74226417202208332339433651144, −4.74674682878995638327710558012, −4.23475498379602944697118359952, −3.09156007584488427452202100240, −1.76703788350375609238217281573, −0.57564484538035185790535747838,
0.875253691115500893813222898812, 2.55520901391792703999781709905, 3.18798985355806012323463135730, 4.26708407797805704491817302333, 5.21272192466764670694753226779, 6.19809357180323068479368311017, 7.02172712580857441115836084026, 7.64388161849244642374893699642, 8.496755826166832043412463374033, 9.287875662559289864715544640810