L(s) = 1 | + (−0.707 + 0.707i)2-s + (0.707 − 0.707i)3-s − 1.00i·4-s + 1.00i·6-s + (−0.189 + 2.63i)7-s + (0.707 + 0.707i)8-s − 1.00i·9-s − 5.46·11-s + (−0.707 − 0.707i)12-s + (2.63 − 2.63i)13-s + (−1.73 − 2i)14-s − 1.00·16-s + (−3.53 − 3.53i)17-s + (0.707 + 0.707i)18-s − 7.46·19-s + ⋯ |
L(s) = 1 | + (−0.499 + 0.499i)2-s + (0.408 − 0.408i)3-s − 0.500i·4-s + 0.408i·6-s + (−0.0716 + 0.997i)7-s + (0.250 + 0.250i)8-s − 0.333i·9-s − 1.64·11-s + (−0.204 − 0.204i)12-s + (0.731 − 0.731i)13-s + (−0.462 − 0.534i)14-s − 0.250·16-s + (−0.857 − 0.857i)17-s + (0.166 + 0.166i)18-s − 1.71·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.663 + 0.747i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.663 + 0.747i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.3507653883\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3507653883\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.707 - 0.707i)T \) |
| 3 | \( 1 + (-0.707 + 0.707i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (0.189 - 2.63i)T \) |
good | 11 | \( 1 + 5.46T + 11T^{2} \) |
| 13 | \( 1 + (-2.63 + 2.63i)T - 13iT^{2} \) |
| 17 | \( 1 + (3.53 + 3.53i)T + 17iT^{2} \) |
| 19 | \( 1 + 7.46T + 19T^{2} \) |
| 23 | \( 1 + (0.707 + 0.707i)T + 23iT^{2} \) |
| 29 | \( 1 + 7.73iT - 29T^{2} \) |
| 31 | \( 1 - 7.92iT - 31T^{2} \) |
| 37 | \( 1 + (-4.89 + 4.89i)T - 37iT^{2} \) |
| 41 | \( 1 + 5.73iT - 41T^{2} \) |
| 43 | \( 1 + (2.63 + 2.63i)T + 43iT^{2} \) |
| 47 | \( 1 + (-5.93 - 5.93i)T + 47iT^{2} \) |
| 53 | \( 1 + (2.50 + 2.50i)T + 53iT^{2} \) |
| 59 | \( 1 + 7.19T + 59T^{2} \) |
| 61 | \( 1 - 2.46iT - 61T^{2} \) |
| 67 | \( 1 + (-4.52 + 4.52i)T - 67iT^{2} \) |
| 71 | \( 1 + 11.4T + 71T^{2} \) |
| 73 | \( 1 + (8.76 - 8.76i)T - 73iT^{2} \) |
| 79 | \( 1 + 12.3iT - 79T^{2} \) |
| 83 | \( 1 + (-0.328 + 0.328i)T - 83iT^{2} \) |
| 89 | \( 1 + 12T + 89T^{2} \) |
| 97 | \( 1 + (-2.17 - 2.17i)T + 97iT^{2} \) |
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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
−9.340909115579301224462306004903, −8.577690273912035427485236357887, −8.142544535331738900428856074371, −7.24812945871652201530015571158, −6.21696756007795980753948863773, −5.58583844584053470081237248366, −4.50170304629671289885172675564, −2.85187801442854934959687113065, −2.14011546880866168301864315056, −0.16267795843312822030465083744,
1.74987907280007489240298717788, 2.85554319197831453240911812308, 4.03132427745053649826209954858, 4.59117044359014462307946810419, 6.08729875402000624266572257839, 7.04758924120078244796961171584, 8.073457939663207696518497821094, 8.494359252991799122010955972784, 9.483264852611221684458418792395, 10.39034536502388869951345287692