| L(s) = 1 | − 0.933i·3-s + (−0.723 − 4.94i)5-s − 6.91·7-s + 8.12·9-s − 11.8·11-s + 13.4·13-s + (−4.61 + 0.675i)15-s − 28.7i·17-s − 30.1·19-s + 6.45i·21-s + 39.5·23-s + (−23.9 + 7.15i)25-s − 15.9i·27-s + 18.9i·29-s + 6.41i·31-s + ⋯ |
| L(s) = 1 | − 0.311i·3-s + (−0.144 − 0.989i)5-s − 0.987·7-s + 0.903·9-s − 1.07·11-s + 1.03·13-s + (−0.307 + 0.0450i)15-s − 1.69i·17-s − 1.58·19-s + 0.307i·21-s + 1.71·23-s + (−0.958 + 0.286i)25-s − 0.592i·27-s + 0.655i·29-s + 0.206i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.801 - 0.597i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.801 - 0.597i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.3768903786\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3768903786\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (0.723 + 4.94i)T \) |
| good | 3 | \( 1 + 0.933iT - 9T^{2} \) |
| 7 | \( 1 + 6.91T + 49T^{2} \) |
| 11 | \( 1 + 11.8T + 121T^{2} \) |
| 13 | \( 1 - 13.4T + 169T^{2} \) |
| 17 | \( 1 + 28.7iT - 289T^{2} \) |
| 19 | \( 1 + 30.1T + 361T^{2} \) |
| 23 | \( 1 - 39.5T + 529T^{2} \) |
| 29 | \( 1 - 18.9iT - 841T^{2} \) |
| 31 | \( 1 - 6.41iT - 961T^{2} \) |
| 37 | \( 1 + 51.5T + 1.36e3T^{2} \) |
| 41 | \( 1 - 17.0T + 1.68e3T^{2} \) |
| 43 | \( 1 + 58.0iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 48.1T + 2.20e3T^{2} \) |
| 53 | \( 1 - 4.02T + 2.80e3T^{2} \) |
| 59 | \( 1 + 45.6T + 3.48e3T^{2} \) |
| 61 | \( 1 - 37.8iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 3.82iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 119. iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 82.9iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 83.2iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 115. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 54.7T + 7.92e3T^{2} \) |
| 97 | \( 1 + 51.8iT - 9.40e3T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.918359713098509759837730090846, −8.339324597061722274529057317539, −7.17132472518611260550756444986, −6.73711258791977213885989718139, −5.49540771408903528024562766073, −4.79441959294330444522220003132, −3.74697366290272456600541488849, −2.63428896186254858525830500055, −1.24272277285490486622083848874, −0.11313021604339822737310627097,
1.80668970008366587801157894140, 3.09693666213785418771730279454, 3.74806201682293639710813494206, 4.74932632433795430296569968469, 6.24226325325976885229697505836, 6.39626341661333626171156823969, 7.49774200483507148916625296569, 8.317545576767948674997523916460, 9.247195502754743748796652013494, 10.23408956895628846679045941557
