L(s) = 1 | + (0.864 + 2.87i)3-s + 9.02i·7-s + (−7.50 + 4.96i)9-s + 21.8i·11-s − 21.6i·13-s + 12.1·17-s − 3.03·19-s + (−25.9 + 7.80i)21-s − 28.5·23-s + (−20.7 − 17.2i)27-s − 12.0i·29-s + 2.19·31-s + (−62.8 + 18.9i)33-s + 0.839i·37-s + (62.2 − 18.7i)39-s + ⋯ |
L(s) = 1 | + (0.288 + 0.957i)3-s + 1.28i·7-s + (−0.833 + 0.551i)9-s + 1.98i·11-s − 1.66i·13-s + 0.712·17-s − 0.159·19-s + (−1.23 + 0.371i)21-s − 1.24·23-s + (−0.768 − 0.639i)27-s − 0.415i·29-s + 0.0706·31-s + (−1.90 + 0.573i)33-s + 0.0227i·37-s + (1.59 − 0.480i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.985 - 0.170i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.985 - 0.170i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.362388408\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.362388408\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.864 - 2.87i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - 9.02iT - 49T^{2} \) |
| 11 | \( 1 - 21.8iT - 121T^{2} \) |
| 13 | \( 1 + 21.6iT - 169T^{2} \) |
| 17 | \( 1 - 12.1T + 289T^{2} \) |
| 19 | \( 1 + 3.03T + 361T^{2} \) |
| 23 | \( 1 + 28.5T + 529T^{2} \) |
| 29 | \( 1 + 12.0iT - 841T^{2} \) |
| 31 | \( 1 - 2.19T + 961T^{2} \) |
| 37 | \( 1 - 0.839iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 35.5iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 12.7iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 22.5T + 2.20e3T^{2} \) |
| 53 | \( 1 - 9.13T + 2.80e3T^{2} \) |
| 59 | \( 1 - 80.4iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 57.8T + 3.72e3T^{2} \) |
| 67 | \( 1 + 63.0iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 17.0iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 52.1iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 7.46T + 6.24e3T^{2} \) |
| 83 | \( 1 + 82.3T + 6.88e3T^{2} \) |
| 89 | \( 1 - 27.5iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 114. iT - 9.40e3T^{2} \) |
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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
−10.55169143121375292305588823535, −9.936905512688399750302172388710, −9.351142116468740586883815122067, −8.244749365174078484610513983348, −7.64471158235923369122383416847, −6.05760159489749202417251309171, −5.28318967136694030390127903018, −4.41149612868895931757962760771, −3.09348690122513284465135293999, −2.13300706632160741970892265784,
0.47926496695637762930472481488, 1.68191178871674747434304211192, 3.25636277820808545451567043277, 4.10251591776792018589232272955, 5.73436229336182931055025889431, 6.54651944729866215455894290238, 7.34393366424451315038782904565, 8.244681494804161383958546992458, 8.939122451108887680623394902966, 10.07774219945286693000317292691