L(s) = 1 | + (1.22 − 1.22i)2-s + (−1.22 − 1.22i)3-s − 0.999i·4-s − 2.99·6-s + (1.22 + 1.22i)8-s + 2.99i·9-s + (−1.22 + 1.22i)12-s + 5·16-s + (−4.89 + 4.89i)17-s + (3.67 + 3.67i)18-s − 4i·19-s + (−2.44 − 2.44i)23-s − 3.00i·24-s + (3.67 − 3.67i)27-s − 8·31-s + (3.67 − 3.67i)32-s + ⋯ |
L(s) = 1 | + (0.866 − 0.866i)2-s + (−0.707 − 0.707i)3-s − 0.499i·4-s − 1.22·6-s + (0.433 + 0.433i)8-s + 0.999i·9-s + (−0.353 + 0.353i)12-s + 1.25·16-s + (−1.18 + 1.18i)17-s + (0.866 + 0.866i)18-s − 0.917i·19-s + (−0.510 − 0.510i)23-s − 0.612i·24-s + (0.707 − 0.707i)27-s − 1.43·31-s + (0.649 − 0.649i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.229 + 0.973i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.229 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.904589 - 0.715908i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.904589 - 0.715908i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (1.22 + 1.22i)T \) |
| 5 | \( 1 \) |
good | 2 | \( 1 + (-1.22 + 1.22i)T - 2iT^{2} \) |
| 7 | \( 1 + 7iT^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 - 13iT^{2} \) |
| 17 | \( 1 + (4.89 - 4.89i)T - 17iT^{2} \) |
| 19 | \( 1 + 4iT - 19T^{2} \) |
| 23 | \( 1 + (2.44 + 2.44i)T + 23iT^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + 8T + 31T^{2} \) |
| 37 | \( 1 + 37iT^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 - 43iT^{2} \) |
| 47 | \( 1 + (-7.34 + 7.34i)T - 47iT^{2} \) |
| 53 | \( 1 + (-9.79 - 9.79i)T + 53iT^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 + 67iT^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 - 73iT^{2} \) |
| 79 | \( 1 - 16iT - 79T^{2} \) |
| 83 | \( 1 + (2.44 + 2.44i)T + 83iT^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 + 97iT^{2} \) |
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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
−13.78849691301066602785506767810, −13.04551560405313689181081413683, −12.27147786681798035208451286844, −11.23859886643624843861479732933, −10.53061872040027379540653170350, −8.509438765201328517434046221591, −7.03391025821551306767908559308, −5.59619503202360186135513635054, −4.21091389250569490413296196158, −2.17607389815220050882504286441,
3.93615513183635762614798187508, 5.11986073900510071403803114577, 6.14944178014170683784485800338, 7.34620001755066011427968295329, 9.229114639916129374277969711062, 10.41457947495543246540209379939, 11.58250579457822642766888248788, 12.79465046856314394413945040041, 13.96125831356930491890603751287, 14.88591915642086371833069085244