L(s) = 1 | + 2.82·5-s − 1.41i·7-s − 2i·13-s − 1.41i·17-s + 2.82·19-s + 6·23-s + 3.00·25-s + 5.65·29-s + 9.89i·31-s − 4.00i·35-s − 2i·37-s − 4.24i·41-s − 8.48·43-s − 10·47-s + 5·49-s + ⋯ |
L(s) = 1 | + 1.26·5-s − 0.534i·7-s − 0.554i·13-s − 0.342i·17-s + 0.648·19-s + 1.25·23-s + 0.600·25-s + 1.05·29-s + 1.77i·31-s − 0.676i·35-s − 0.328i·37-s − 0.662i·41-s − 1.29·43-s − 1.45·47-s + 0.714·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4608 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.816 + 0.577i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4608 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.816 + 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.661150393\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.661150393\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 - 2.82T + 5T^{2} \) |
| 7 | \( 1 + 1.41iT - 7T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 + 2iT - 13T^{2} \) |
| 17 | \( 1 + 1.41iT - 17T^{2} \) |
| 19 | \( 1 - 2.82T + 19T^{2} \) |
| 23 | \( 1 - 6T + 23T^{2} \) |
| 29 | \( 1 - 5.65T + 29T^{2} \) |
| 31 | \( 1 - 9.89iT - 31T^{2} \) |
| 37 | \( 1 + 2iT - 37T^{2} \) |
| 41 | \( 1 + 4.24iT - 41T^{2} \) |
| 43 | \( 1 + 8.48T + 43T^{2} \) |
| 47 | \( 1 + 10T + 47T^{2} \) |
| 53 | \( 1 - 5.65T + 53T^{2} \) |
| 59 | \( 1 - 12iT - 59T^{2} \) |
| 61 | \( 1 + 10iT - 61T^{2} \) |
| 67 | \( 1 - 5.65T + 67T^{2} \) |
| 71 | \( 1 - 2T + 71T^{2} \) |
| 73 | \( 1 - 10T + 73T^{2} \) |
| 79 | \( 1 + 12.7iT - 79T^{2} \) |
| 83 | \( 1 + 4iT - 83T^{2} \) |
| 89 | \( 1 + 9.89iT - 89T^{2} \) |
| 97 | \( 1 - 12T + 97T^{2} \) |
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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
−8.380761947456922760715770054216, −7.34028740966495056505085870907, −6.82969308886975178924486780603, −6.07354927287348495114561786600, −5.18869902695822134010980756291, −4.84353114784319437315436299077, −3.48622815491865014169050392003, −2.84663025818247163547277017920, −1.75495457466366195331429611955, −0.838369058726951553592484173434,
1.10001293584377675054322218066, 2.09108274125028759425641351592, 2.76794760390716487654127671663, 3.81739057146245058494164130206, 4.96057481412867214042343804655, 5.37386578189162678006975069305, 6.35269923617648039985100612097, 6.62492686740849310730390666309, 7.72102631175372755554800593861, 8.500017587406057626792191331206