| L(s) = 1 | + (1.22 − 1.22i)3-s + (3.67 + 3.67i)7-s − 2.99i·9-s + (−1.22 + 1.22i)13-s − 7i·19-s + 9·21-s + (−3.67 − 3.67i)27-s − 11·31-s + (4.89 + 4.89i)37-s + 2.99i·39-s + (−1.22 + 1.22i)43-s + 20i·49-s + (−8.57 − 8.57i)57-s − 61-s + (11.0 − 11.0i)63-s + ⋯ |
| L(s) = 1 | + (0.707 − 0.707i)3-s + (1.38 + 1.38i)7-s − 0.999i·9-s + (−0.339 + 0.339i)13-s − 1.60i·19-s + 1.96·21-s + (−0.707 − 0.707i)27-s − 1.97·31-s + (0.805 + 0.805i)37-s + 0.480i·39-s + (−0.186 + 0.186i)43-s + 2.85i·49-s + (−1.13 − 1.13i)57-s − 0.128·61-s + (1.38 − 1.38i)63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.945 + 0.326i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.945 + 0.326i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.73555 - 0.291225i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.73555 - 0.291225i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.22 + 1.22i)T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + (-3.67 - 3.67i)T + 7iT^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 + (1.22 - 1.22i)T - 13iT^{2} \) |
| 17 | \( 1 - 17iT^{2} \) |
| 19 | \( 1 + 7iT - 19T^{2} \) |
| 23 | \( 1 + 23iT^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + 11T + 31T^{2} \) |
| 37 | \( 1 + (-4.89 - 4.89i)T + 37iT^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 + (1.22 - 1.22i)T - 43iT^{2} \) |
| 47 | \( 1 - 47iT^{2} \) |
| 53 | \( 1 + 53iT^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + T + 61T^{2} \) |
| 67 | \( 1 + (8.57 + 8.57i)T + 67iT^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 + (9.79 - 9.79i)T - 73iT^{2} \) |
| 79 | \( 1 - 4iT - 79T^{2} \) |
| 83 | \( 1 + 83iT^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 + (-3.67 - 3.67i)T + 97iT^{2} \) |
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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
−11.72877412032451963904592078077, −11.09738498905289524772231121769, −9.377872132847531494493509109494, −8.822555355197016147019460866317, −7.953120423714196314595464752976, −7.00924955764891364215982017054, −5.71177491798444827396276226959, −4.60886312772752862324915921247, −2.80015991950373825266521744282, −1.78628838782571669735172043363,
1.77973688557852886147883827335, 3.61653648692917148728047209142, 4.44020035648509998984483148507, 5.52482853605190452122363885586, 7.43842888431618224300548340974, 7.84148658730117904538530503067, 8.915701980423007964693690494091, 10.13113842523790959479490727842, 10.64422132286676405041391494598, 11.52030785592762076557666464599
