L(s) = 1 | + (0.707 − 0.707i)2-s − 1.00i·4-s + (−2 + i)5-s + (2 + 2i)7-s + (−0.707 − 0.707i)8-s + (−0.707 + 2.12i)10-s − 0.828i·11-s + (1.41 − 1.41i)13-s + 2.82·14-s − 1.00·16-s + (−5.41 + 5.41i)17-s + 3.41i·19-s + (1.00 + 2.00i)20-s + (−0.585 − 0.585i)22-s + (−0.707 − 0.707i)23-s + ⋯ |
L(s) = 1 | + (0.499 − 0.499i)2-s − 0.500i·4-s + (−0.894 + 0.447i)5-s + (0.755 + 0.755i)7-s + (−0.250 − 0.250i)8-s + (−0.223 + 0.670i)10-s − 0.249i·11-s + (0.392 − 0.392i)13-s + 0.755·14-s − 0.250·16-s + (−1.31 + 1.31i)17-s + 0.783i·19-s + (0.223 + 0.447i)20-s + (−0.124 − 0.124i)22-s + (−0.147 − 0.147i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0618 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0618 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.132274076\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.132274076\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 + (2 - i)T \) |
| 23 | \( 1 + (0.707 + 0.707i)T \) |
good | 7 | \( 1 + (-2 - 2i)T + 7iT^{2} \) |
| 11 | \( 1 + 0.828iT - 11T^{2} \) |
| 13 | \( 1 + (-1.41 + 1.41i)T - 13iT^{2} \) |
| 17 | \( 1 + (5.41 - 5.41i)T - 17iT^{2} \) |
| 19 | \( 1 - 3.41iT - 19T^{2} \) |
| 29 | \( 1 + 4.82T + 29T^{2} \) |
| 31 | \( 1 + 4.82T + 31T^{2} \) |
| 37 | \( 1 + (-1.41 - 1.41i)T + 37iT^{2} \) |
| 41 | \( 1 - 2.58iT - 41T^{2} \) |
| 43 | \( 1 + (6.24 - 6.24i)T - 43iT^{2} \) |
| 47 | \( 1 - 47iT^{2} \) |
| 53 | \( 1 + (-2.65 - 2.65i)T + 53iT^{2} \) |
| 59 | \( 1 + 2.82T + 59T^{2} \) |
| 61 | \( 1 - 2.58T + 61T^{2} \) |
| 67 | \( 1 + (-1.17 - 1.17i)T + 67iT^{2} \) |
| 71 | \( 1 - 4.58iT - 71T^{2} \) |
| 73 | \( 1 + (4.17 - 4.17i)T - 73iT^{2} \) |
| 79 | \( 1 - 6.82iT - 79T^{2} \) |
| 83 | \( 1 + (-9.24 - 9.24i)T + 83iT^{2} \) |
| 89 | \( 1 + 14T + 89T^{2} \) |
| 97 | \( 1 + (-5.07 - 5.07i)T + 97iT^{2} \) |
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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
−9.285518643942952075420614120927, −8.347937908396669909611188454876, −8.059631380329905412059222381256, −6.84708099032614032862067955289, −6.05434672796665933264864644962, −5.26854521204282197317525119259, −4.22477272881341768399841202140, −3.65276337448274947310267487633, −2.55388651689387236224425188353, −1.56197511301312986001106556733,
0.32933656310688977517867744759, 1.92762143685178632549701404646, 3.33736458975109943476046577570, 4.28748803479408918212061833987, 4.68627655419002978057754615303, 5.54712496281925747459358854830, 6.92603814497777779355044381181, 7.17572562052806203411508789526, 7.984228220253664236191941149895, 8.834914300244332835871732289934