L(s) = 1 | − i·3-s + (2.20 − 0.345i)5-s + 0.0620i·7-s − 9-s + 3.73·11-s + 3.61i·13-s + (−0.345 − 2.20i)15-s + 5.75i·17-s − 4.84·19-s + 0.0620·21-s − 3.34i·23-s + (4.76 − 1.52i)25-s + i·27-s − 4.90·29-s + 9.28·31-s + ⋯ |
L(s) = 1 | − 0.577i·3-s + (0.988 − 0.154i)5-s + 0.0234i·7-s − 0.333·9-s + 1.12·11-s + 1.00i·13-s + (−0.0891 − 0.570i)15-s + 1.39i·17-s − 1.11·19-s + 0.0135·21-s − 0.697i·23-s + (0.952 − 0.305i)25-s + 0.192i·27-s − 0.910·29-s + 1.66·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.988 - 0.154i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.988 - 0.154i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.416561974\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.416561974\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + iT \) |
| 5 | \( 1 + (-2.20 + 0.345i)T \) |
| 67 | \( 1 + iT \) |
good | 7 | \( 1 - 0.0620iT - 7T^{2} \) |
| 11 | \( 1 - 3.73T + 11T^{2} \) |
| 13 | \( 1 - 3.61iT - 13T^{2} \) |
| 17 | \( 1 - 5.75iT - 17T^{2} \) |
| 19 | \( 1 + 4.84T + 19T^{2} \) |
| 23 | \( 1 + 3.34iT - 23T^{2} \) |
| 29 | \( 1 + 4.90T + 29T^{2} \) |
| 31 | \( 1 - 9.28T + 31T^{2} \) |
| 37 | \( 1 + 3.35iT - 37T^{2} \) |
| 41 | \( 1 - 9.38T + 41T^{2} \) |
| 43 | \( 1 - 2.89iT - 43T^{2} \) |
| 47 | \( 1 - 10.5iT - 47T^{2} \) |
| 53 | \( 1 - 9.91iT - 53T^{2} \) |
| 59 | \( 1 + 9.42T + 59T^{2} \) |
| 61 | \( 1 + 11.5T + 61T^{2} \) |
| 71 | \( 1 - 14.9T + 71T^{2} \) |
| 73 | \( 1 - 10.5iT - 73T^{2} \) |
| 79 | \( 1 - 13.4T + 79T^{2} \) |
| 83 | \( 1 + 3.66iT - 83T^{2} \) |
| 89 | \( 1 + 5.66T + 89T^{2} \) |
| 97 | \( 1 + 2.89iT - 97T^{2} \) |
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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
−8.565179517846459469002425818160, −7.76131797393681556183244443532, −6.73638320514817470886906614037, −6.22850772220549624960794267559, −5.90947113263030741513243421838, −4.52029446289201229906857847443, −4.07919527263081636049081555679, −2.68864475236152672284152887598, −1.88747146076460332176074234130, −1.14382089998777796317196624649,
0.76764151334434437945647322767, 2.05782247597768988870067680679, 2.93660809426151574227454634413, 3.79606364259952775963160079847, 4.75956196298751255717801297964, 5.41851529003629142194559079706, 6.20276568574750059775742351578, 6.78617576348971694655592846985, 7.72721381180599663533308347687, 8.599872744292506802296082685051