L(s) = 1 | + i·2-s + (−0.0866 + 1.72i)3-s − 4-s − 0.713·5-s + (−1.72 − 0.0866i)6-s + i·7-s − i·8-s + (−2.98 − 0.299i)9-s − 0.713i·10-s − 2.60·11-s + (0.0866 − 1.72i)12-s − 5.59·13-s − 14-s + (0.0618 − 1.23i)15-s + 16-s + 6.17·17-s + ⋯ |
L(s) = 1 | + 0.707i·2-s + (−0.0500 + 0.998i)3-s − 0.5·4-s − 0.319·5-s + (−0.706 − 0.0353i)6-s + 0.377i·7-s − 0.353i·8-s + (−0.994 − 0.0999i)9-s − 0.225i·10-s − 0.784·11-s + (0.0250 − 0.499i)12-s − 1.55·13-s − 0.267·14-s + (0.0159 − 0.318i)15-s + 0.250·16-s + 1.49·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.312 + 0.949i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.312 + 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0160699 - 0.0116272i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0160699 - 0.0116272i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (0.0866 - 1.72i)T \) |
| 7 | \( 1 - iT \) |
| 23 | \( 1 + (-4.47 + 1.72i)T \) |
good | 5 | \( 1 + 0.713T + 5T^{2} \) |
| 11 | \( 1 + 2.60T + 11T^{2} \) |
| 13 | \( 1 + 5.59T + 13T^{2} \) |
| 17 | \( 1 - 6.17T + 17T^{2} \) |
| 19 | \( 1 + 3.14iT - 19T^{2} \) |
| 29 | \( 1 + 0.797iT - 29T^{2} \) |
| 31 | \( 1 + 7.70T + 31T^{2} \) |
| 37 | \( 1 + 7.21iT - 37T^{2} \) |
| 41 | \( 1 - 3.84iT - 41T^{2} \) |
| 43 | \( 1 - 4.89iT - 43T^{2} \) |
| 47 | \( 1 - 4.60iT - 47T^{2} \) |
| 53 | \( 1 + 13.4T + 53T^{2} \) |
| 59 | \( 1 + 11.2iT - 59T^{2} \) |
| 61 | \( 1 - 1.01iT - 61T^{2} \) |
| 67 | \( 1 + 13.9iT - 67T^{2} \) |
| 71 | \( 1 - 4.29iT - 71T^{2} \) |
| 73 | \( 1 - 7.05T + 73T^{2} \) |
| 79 | \( 1 + 5.48iT - 79T^{2} \) |
| 83 | \( 1 - 10.1T + 83T^{2} \) |
| 89 | \( 1 + 8.77T + 89T^{2} \) |
| 97 | \( 1 - 14.9iT - 97T^{2} \) |
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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
−9.523786523048060467670563564221, −9.312668777705644337878552295160, −7.945146265571028054290110939420, −7.61529974181824105819874857160, −6.30949726853847777104917346977, −5.17535389576878772331201983391, −4.97102686044606758314397089747, −3.64066595693381087686524839128, −2.63159639135062991190859728374, −0.009157201714916155286619684896,
1.48969316750350047959724295975, 2.68137841960483304304116778965, 3.60586490653732825517626765205, 5.05247486961146435732447727038, 5.68275583392166208223894812127, 7.16658530269746220701483659954, 7.60069396899702875451327307854, 8.356451332725971527165387342109, 9.549245803027019830077994220830, 10.21877513034978290725967756643