| L(s) = 1 | + (2 + 2i)3-s + (−1 + 2i)5-s + (−2 − 2i)7-s + 5i·9-s + (1 − i)13-s + (−6 + 2i)15-s + (−5 + 5i)17-s + 4i·19-s − 8i·21-s + (−2 + 2i)23-s + (−3 − 4i)25-s + (−4 + 4i)27-s − 4·29-s + 4i·31-s + (6 − 2i)35-s + ⋯ |
| L(s) = 1 | + (1.15 + 1.15i)3-s + (−0.447 + 0.894i)5-s + (−0.755 − 0.755i)7-s + 1.66i·9-s + (0.277 − 0.277i)13-s + (−1.54 + 0.516i)15-s + (−1.21 + 1.21i)17-s + 0.917i·19-s − 1.74i·21-s + (−0.417 + 0.417i)23-s + (−0.600 − 0.800i)25-s + (−0.769 + 0.769i)27-s − 0.742·29-s + 0.718i·31-s + (1.01 − 0.338i)35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.973 - 0.229i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.973 - 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.441969078\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.441969078\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (1 - 2i)T \) |
| good | 3 | \( 1 + (-2 - 2i)T + 3iT^{2} \) |
| 7 | \( 1 + (2 + 2i)T + 7iT^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + (-1 + i)T - 13iT^{2} \) |
| 17 | \( 1 + (5 - 5i)T - 17iT^{2} \) |
| 19 | \( 1 - 4iT - 19T^{2} \) |
| 23 | \( 1 + (2 - 2i)T - 23iT^{2} \) |
| 29 | \( 1 + 4T + 29T^{2} \) |
| 31 | \( 1 - 4iT - 31T^{2} \) |
| 37 | \( 1 + (1 + i)T + 37iT^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 + (-6 - 6i)T + 43iT^{2} \) |
| 47 | \( 1 + (2 + 2i)T + 47iT^{2} \) |
| 53 | \( 1 + (7 - 7i)T - 53iT^{2} \) |
| 59 | \( 1 + 4iT - 59T^{2} \) |
| 61 | \( 1 + 4iT - 61T^{2} \) |
| 67 | \( 1 + (-10 + 10i)T - 67iT^{2} \) |
| 71 | \( 1 - 12iT - 71T^{2} \) |
| 73 | \( 1 + (-3 - 3i)T + 73iT^{2} \) |
| 79 | \( 1 - 16T + 79T^{2} \) |
| 83 | \( 1 + (2 + 2i)T + 83iT^{2} \) |
| 89 | \( 1 - 89T^{2} \) |
| 97 | \( 1 + (3 - 3i)T - 97iT^{2} \) |
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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
−9.973679911939139526748356336571, −9.413182050107134106519429533585, −8.385563852228643003862097865187, −7.85023832529147102938108374188, −6.83899886389135080126989629111, −5.99107627650993733301132069094, −4.50204448171704456851054176345, −3.68452738103680850149415273790, −3.40358533782168554823678821481, −2.14672615844771159328365190354,
0.48801766974882057292402732214, 2.02087816636786727949629038252, 2.77130085051307419705626505132, 3.88711225932820422659700164371, 5.00700533343344961201314165871, 6.21933245936249948878567538732, 6.97863353916608501655469369985, 7.72886987944764369103945800197, 8.584297267753593967619216433470, 9.126736827397984662897507742928
