L(s) = 1 | − i·2-s + (−0.707 + 0.707i)3-s − 4-s + (1.71 + 1.43i)5-s + (0.707 + 0.707i)6-s + (3.31 − 3.31i)7-s + i·8-s − 1.00i·9-s + (1.43 − 1.71i)10-s − 4.55i·11-s + (0.707 − 0.707i)12-s − 0.0409i·13-s + (−3.31 − 3.31i)14-s + (−2.22 + 0.199i)15-s + 16-s − 3.39·17-s + ⋯ |
L(s) = 1 | − 0.707i·2-s + (−0.408 + 0.408i)3-s − 0.5·4-s + (0.767 + 0.641i)5-s + (0.288 + 0.288i)6-s + (1.25 − 1.25i)7-s + 0.353i·8-s − 0.333i·9-s + (0.453 − 0.542i)10-s − 1.37i·11-s + (0.204 − 0.204i)12-s − 0.0113i·13-s + (−0.885 − 0.885i)14-s + (−0.575 + 0.0514i)15-s + 0.250·16-s − 0.822·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0183 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0183 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.617874307\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.617874307\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + iT \) |
| 3 | \( 1 + (0.707 - 0.707i)T \) |
| 5 | \( 1 + (-1.71 - 1.43i)T \) |
| 37 | \( 1 + (1.32 - 5.93i)T \) |
good | 7 | \( 1 + (-3.31 + 3.31i)T - 7iT^{2} \) |
| 11 | \( 1 + 4.55iT - 11T^{2} \) |
| 13 | \( 1 + 0.0409iT - 13T^{2} \) |
| 17 | \( 1 + 3.39T + 17T^{2} \) |
| 19 | \( 1 + (1.42 - 1.42i)T - 19iT^{2} \) |
| 23 | \( 1 + 6.74iT - 23T^{2} \) |
| 29 | \( 1 + (-2.73 - 2.73i)T + 29iT^{2} \) |
| 31 | \( 1 + (-2.17 + 2.17i)T - 31iT^{2} \) |
| 41 | \( 1 + 4.17iT - 41T^{2} \) |
| 43 | \( 1 + 8.29iT - 43T^{2} \) |
| 47 | \( 1 + (-1.79 + 1.79i)T - 47iT^{2} \) |
| 53 | \( 1 + (5.35 + 5.35i)T + 53iT^{2} \) |
| 59 | \( 1 + (7.59 - 7.59i)T - 59iT^{2} \) |
| 61 | \( 1 + (-4.71 + 4.71i)T - 61iT^{2} \) |
| 67 | \( 1 + (-10.3 - 10.3i)T + 67iT^{2} \) |
| 71 | \( 1 - 3.26T + 71T^{2} \) |
| 73 | \( 1 + (-3.99 + 3.99i)T - 73iT^{2} \) |
| 79 | \( 1 + (3.34 - 3.34i)T - 79iT^{2} \) |
| 83 | \( 1 + (-1.21 - 1.21i)T + 83iT^{2} \) |
| 89 | \( 1 + (9.81 + 9.81i)T + 89iT^{2} \) |
| 97 | \( 1 - 16.9T + 97T^{2} \) |
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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
−10.15308696647593337337604698057, −8.822827822449334062601083264016, −8.271537496088439552318760407243, −7.02828030195235088813515521029, −6.20071722497390909130528649473, −5.15343077207722144359781034223, −4.33907619254214670946764853685, −3.39314326558217559472396905810, −2.12116018985054768111219748631, −0.801970490310645066243087810967,
1.54315623714510599465816372360, 2.34015244578910255028165043151, 4.57050488201185187138353711157, 4.95303217069590826367471600284, 5.80013875134754393383754688497, 6.56330287248879324104013070039, 7.63254731640293747795087698803, 8.304936068687294024075238802658, 9.172490949833814591903100045080, 9.691454870674575582182528314180