| L(s) = 1 | − 2.72i·2-s − 3.44·4-s − 5.71·5-s + 8.69·7-s − 1.52i·8-s + 15.5i·10-s − 6.01i·11-s − 4.81i·13-s − 23.7i·14-s − 17.9·16-s − 10.8·17-s − 18.4·19-s + 19.6·20-s − 16.4·22-s − 15.4i·23-s + ⋯ |
| L(s) = 1 | − 1.36i·2-s − 0.860·4-s − 1.14·5-s + 1.24·7-s − 0.190i·8-s + 1.55i·10-s − 0.546i·11-s − 0.370i·13-s − 1.69i·14-s − 1.12·16-s − 0.637·17-s − 0.971·19-s + 0.983·20-s − 0.745·22-s − 0.669i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.499 - 0.866i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.499 - 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.7226207108\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7226207108\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 59 | \( 1 + (29.4 + 51.1i)T \) |
| good | 2 | \( 1 + 2.72iT - 4T^{2} \) |
| 5 | \( 1 + 5.71T + 25T^{2} \) |
| 7 | \( 1 - 8.69T + 49T^{2} \) |
| 11 | \( 1 + 6.01iT - 121T^{2} \) |
| 13 | \( 1 + 4.81iT - 169T^{2} \) |
| 17 | \( 1 + 10.8T + 289T^{2} \) |
| 19 | \( 1 + 18.4T + 361T^{2} \) |
| 23 | \( 1 + 15.4iT - 529T^{2} \) |
| 29 | \( 1 + 15.5T + 841T^{2} \) |
| 31 | \( 1 + 6.38iT - 961T^{2} \) |
| 37 | \( 1 + 0.527iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 9.31T + 1.68e3T^{2} \) |
| 43 | \( 1 - 52.9iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 78.0iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 10.2T + 2.80e3T^{2} \) |
| 61 | \( 1 + 41.5iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 92.4iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 93.8T + 5.04e3T^{2} \) |
| 73 | \( 1 + 88.0iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 81.2T + 6.24e3T^{2} \) |
| 83 | \( 1 - 2.92iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 168. iT - 7.92e3T^{2} \) |
| 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
−10.53062662218888784366228806087, −9.249009998700777184748538559081, −8.327197318275424814821718184826, −7.67419028611798435566114549551, −6.35251921250241778969711534099, −4.73898021263564338413228398607, −4.10967234530100619846901395518, −2.97486918082534104786079432430, −1.71224936750579279415223993992, −0.27620043821366476036262211995,
2.01503268752971861152442644064, 4.04136085845865852923299779426, 4.72843613525721265340475824669, 5.73979164082500926852922643058, 7.05156997790824063659018881036, 7.43056528664087481028075604017, 8.415122207615230804874409415219, 8.826810759546470369735149302697, 10.39083458350059271750832619872, 11.47801083043817944802008061858
