| L(s) = 1 | + (1.22 − 1.22i)3-s + (4.77 + 4.77i)7-s − 2.99i·9-s − 20.6·11-s + (8.32 − 8.32i)13-s + (1.34 + 1.34i)17-s + 33.6i·19-s + 11.6·21-s + (−16.0 + 16.0i)23-s + (−3.67 − 3.67i)27-s + 8.69i·29-s + 49.0·31-s + (−25.3 + 25.3i)33-s + (36.4 + 36.4i)37-s − 20.3i·39-s + ⋯ |
| L(s) = 1 | + (0.408 − 0.408i)3-s + (0.682 + 0.682i)7-s − 0.333i·9-s − 1.88·11-s + (0.640 − 0.640i)13-s + (0.0793 + 0.0793i)17-s + 1.77i·19-s + 0.556·21-s + (−0.697 + 0.697i)23-s + (−0.136 − 0.136i)27-s + 0.299i·29-s + 1.58·31-s + (−0.768 + 0.768i)33-s + (0.986 + 0.986i)37-s − 0.522i·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.326 - 0.945i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.326 - 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.749532684\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.749532684\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.22 + 1.22i)T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + (-4.77 - 4.77i)T + 49iT^{2} \) |
| 11 | \( 1 + 20.6T + 121T^{2} \) |
| 13 | \( 1 + (-8.32 + 8.32i)T - 169iT^{2} \) |
| 17 | \( 1 + (-1.34 - 1.34i)T + 289iT^{2} \) |
| 19 | \( 1 - 33.6iT - 361T^{2} \) |
| 23 | \( 1 + (16.0 - 16.0i)T - 529iT^{2} \) |
| 29 | \( 1 - 8.69iT - 841T^{2} \) |
| 31 | \( 1 - 49.0T + 961T^{2} \) |
| 37 | \( 1 + (-36.4 - 36.4i)T + 1.36e3iT^{2} \) |
| 41 | \( 1 - 33.3T + 1.68e3T^{2} \) |
| 43 | \( 1 + (24.1 - 24.1i)T - 1.84e3iT^{2} \) |
| 47 | \( 1 + (13.9 + 13.9i)T + 2.20e3iT^{2} \) |
| 53 | \( 1 + (59.3 - 59.3i)T - 2.80e3iT^{2} \) |
| 59 | \( 1 - 30iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 47.7T + 3.72e3T^{2} \) |
| 67 | \( 1 + (-17.0 - 17.0i)T + 4.48e3iT^{2} \) |
| 71 | \( 1 - 9.30T + 5.04e3T^{2} \) |
| 73 | \( 1 + (22.2 - 22.2i)T - 5.32e3iT^{2} \) |
| 79 | \( 1 + 27.3iT - 6.24e3T^{2} \) |
| 83 | \( 1 + (63.4 - 63.4i)T - 6.88e3iT^{2} \) |
| 89 | \( 1 - 17.3iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (41.8 + 41.8i)T + 9.40e3iT^{2} \) |
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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
−9.855452856464278075278792444614, −8.578023147584300474472008201834, −7.936569792917960957090498465459, −7.76950268649223874506672780177, −6.17083204078803232568139894073, −5.62954624808088068899789018115, −4.65223401450562193236547567918, −3.30760494315131761246799542378, −2.45254583355364213559014455058, −1.32707819818830284044071281363,
0.49468915295161580528199684779, 2.16099170220949301263563291237, 3.03964902260561882728975641963, 4.43405315298708943616371734190, 4.77894314175557407919985425430, 5.98526892603589423436870097890, 7.08912835820192437878670240979, 7.927945427795179836801456260233, 8.424348132213703850185599705603, 9.444313156246940177542181028003
