| L(s) = 1 | + (1.29 − 2.70i)3-s + (1.73 − 3.00i)7-s + (−5.66 − 6.99i)9-s + (−12.3 − 7.14i)11-s + (3.04 + 5.28i)13-s − 11.7i·17-s + 23.4·19-s + (−5.88 − 8.56i)21-s + (−24.8 + 14.3i)23-s + (−26.2 + 6.30i)27-s + (−13.3 − 7.68i)29-s + (−18.4 − 32.0i)31-s + (−35.3 + 24.2i)33-s − 46.7·37-s + (18.2 − 1.43i)39-s + ⋯ |
| L(s) = 1 | + (0.430 − 0.902i)3-s + (0.247 − 0.428i)7-s + (−0.629 − 0.777i)9-s + (−1.12 − 0.649i)11-s + (0.234 + 0.406i)13-s − 0.690i·17-s + 1.23·19-s + (−0.280 − 0.407i)21-s + (−1.08 + 0.624i)23-s + (−0.972 + 0.233i)27-s + (−0.459 − 0.265i)29-s + (−0.596 − 1.03i)31-s + (−1.07 + 0.735i)33-s − 1.26·37-s + (0.467 − 0.0368i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0174i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.999 - 0.0174i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.102698648\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.102698648\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.29 + 2.70i)T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + (-1.73 + 3.00i)T + (-24.5 - 42.4i)T^{2} \) |
| 11 | \( 1 + (12.3 + 7.14i)T + (60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + (-3.04 - 5.28i)T + (-84.5 + 146. i)T^{2} \) |
| 17 | \( 1 + 11.7iT - 289T^{2} \) |
| 19 | \( 1 - 23.4T + 361T^{2} \) |
| 23 | \( 1 + (24.8 - 14.3i)T + (264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + (13.3 + 7.68i)T + (420.5 + 728. i)T^{2} \) |
| 31 | \( 1 + (18.4 + 32.0i)T + (-480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + 46.7T + 1.36e3T^{2} \) |
| 41 | \( 1 + (17.1 - 9.89i)T + (840.5 - 1.45e3i)T^{2} \) |
| 43 | \( 1 + (2.75 - 4.76i)T + (-924.5 - 1.60e3i)T^{2} \) |
| 47 | \( 1 + (-8.93 - 5.15i)T + (1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 + 41.5iT - 2.80e3T^{2} \) |
| 59 | \( 1 + (-58.5 + 33.7i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (26.5 - 45.9i)T + (-1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-15.9 - 27.5i)T + (-2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 - 2.28iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 86.2T + 5.32e3T^{2} \) |
| 79 | \( 1 + (-58.4 + 101. i)T + (-3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + (113. + 65.2i)T + (3.44e3 + 5.96e3i)T^{2} \) |
| 89 | \( 1 + 75.3iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (14.1 - 24.4i)T + (-4.70e3 - 8.14e3i)T^{2} \) |
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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
−9.402796280695097594432494127130, −8.496023134192120152845849944739, −7.65745286897054641969434198784, −7.24427183529916107475711151461, −6.01337712589362372263138037886, −5.27323548148667666196002383684, −3.82004366282573119864598125649, −2.85379961822629903002184170299, −1.67408714234644353766041444714, −0.31710318448268431190429129392,
1.94307845153399594847881190622, 3.02451981450512634234759702601, 4.00972346647718118147769153926, 5.18265907819246802183108503407, 5.58439575568040949028526436175, 7.09842642900742241155701227016, 8.052277945540412338671785838971, 8.609962125138634822930395836247, 9.586094778361859217373440051751, 10.34694157484130629895002504120
