L(s) = 1 | − i·2-s + 1.79·3-s − 4-s + 0.791i·5-s − 1.79i·6-s − 2·7-s + i·8-s + 0.208·9-s + 0.791·10-s − 0.791·11-s − 1.79·12-s − 3.79i·13-s + 2i·14-s + 1.41i·15-s + 16-s + 7.58i·17-s + ⋯ |
L(s) = 1 | − 0.707i·2-s + 1.03·3-s − 0.5·4-s + 0.353i·5-s − 0.731i·6-s − 0.755·7-s + 0.353i·8-s + 0.0695·9-s + 0.250·10-s − 0.238·11-s − 0.517·12-s − 1.05i·13-s + 0.534i·14-s + 0.365i·15-s + 0.250·16-s + 1.83i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.753 + 0.657i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.753 + 0.657i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.01092 - 0.379142i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.01092 - 0.379142i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + iT \) |
| 37 | \( 1 + (-4 + 4.58i)T \) |
good | 3 | \( 1 - 1.79T + 3T^{2} \) |
| 5 | \( 1 - 0.791iT - 5T^{2} \) |
| 7 | \( 1 + 2T + 7T^{2} \) |
| 11 | \( 1 + 0.791T + 11T^{2} \) |
| 13 | \( 1 + 3.79iT - 13T^{2} \) |
| 17 | \( 1 - 7.58iT - 17T^{2} \) |
| 19 | \( 1 - 1.58iT - 19T^{2} \) |
| 23 | \( 1 + 3.79iT - 23T^{2} \) |
| 29 | \( 1 - 3.79iT - 29T^{2} \) |
| 31 | \( 1 + 8.37iT - 31T^{2} \) |
| 41 | \( 1 - 9.79T + 41T^{2} \) |
| 43 | \( 1 + 6iT - 43T^{2} \) |
| 47 | \( 1 + 7.58T + 47T^{2} \) |
| 53 | \( 1 + 1.58T + 53T^{2} \) |
| 59 | \( 1 - 1.58iT - 59T^{2} \) |
| 61 | \( 1 - 12.7iT - 61T^{2} \) |
| 67 | \( 1 - 6.37T + 67T^{2} \) |
| 71 | \( 1 + 9.16T + 71T^{2} \) |
| 73 | \( 1 + 4.37T + 73T^{2} \) |
| 79 | \( 1 - 8.20iT - 79T^{2} \) |
| 83 | \( 1 - 15.1T + 83T^{2} \) |
| 89 | \( 1 - 6iT - 89T^{2} \) |
| 97 | \( 1 + 13.5iT - 97T^{2} \) |
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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
−14.51857667129917132603272307751, −13.14215444081536577234990046141, −12.64020750102322277673949082459, −10.91633864637183167844453857725, −10.04927674140541890680663199485, −8.822974507471760822583719598981, −7.84117264190993255120497014889, −5.97832402100446085088432750213, −3.77761671350164625409554474866, −2.62509516017638999425379875924,
3.04893253332420279706449546079, 4.84351231959469134717944299118, 6.59542030368139108942676692192, 7.78092525267977142108962169786, 9.079728672022026253730412723331, 9.566375257743002105551904355025, 11.53096802022679387544575129822, 12.98493668412561973418339308322, 13.83156345533807426231772609100, 14.57429349606826310858075089623