L(s) = 1 | − 2-s + 4-s + (1.88 − 1.20i)5-s − 8-s + (−1.88 + 1.20i)10-s − 1.59i·11-s − 0.925·13-s + 16-s − 3.95i·17-s − 0.625i·19-s + (1.88 − 1.20i)20-s + 1.59i·22-s + 7.78·23-s + (2.08 − 4.54i)25-s + 0.925·26-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.5·4-s + (0.841 − 0.539i)5-s − 0.353·8-s + (−0.595 + 0.381i)10-s − 0.480i·11-s − 0.256·13-s + 0.250·16-s − 0.959i·17-s − 0.143i·19-s + (0.420 − 0.269i)20-s + 0.339i·22-s + 1.62·23-s + (0.416 − 0.908i)25-s + 0.181·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.619 + 0.784i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.619 + 0.784i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.645335216\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.645335216\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-1.88 + 1.20i)T \) |
| 7 | \( 1 \) |
good | 11 | \( 1 + 1.59iT - 11T^{2} \) |
| 13 | \( 1 + 0.925T + 13T^{2} \) |
| 17 | \( 1 + 3.95iT - 17T^{2} \) |
| 19 | \( 1 + 0.625iT - 19T^{2} \) |
| 23 | \( 1 - 7.78T + 23T^{2} \) |
| 29 | \( 1 - 9.34iT - 29T^{2} \) |
| 31 | \( 1 - 10.3iT - 31T^{2} \) |
| 37 | \( 1 + 0.426iT - 37T^{2} \) |
| 41 | \( 1 - 8.35T + 41T^{2} \) |
| 43 | \( 1 + 6.27iT - 43T^{2} \) |
| 47 | \( 1 - 2.78iT - 47T^{2} \) |
| 53 | \( 1 + 3.35T + 53T^{2} \) |
| 59 | \( 1 - 6.21T + 59T^{2} \) |
| 61 | \( 1 + 11.0iT - 61T^{2} \) |
| 67 | \( 1 - 0.356iT - 67T^{2} \) |
| 71 | \( 1 + 9.07iT - 71T^{2} \) |
| 73 | \( 1 + 6.83T + 73T^{2} \) |
| 79 | \( 1 - 9.05T + 79T^{2} \) |
| 83 | \( 1 + 0.809iT - 83T^{2} \) |
| 89 | \( 1 - 4.01T + 89T^{2} \) |
| 97 | \( 1 + 7.87T + 97T^{2} \) |
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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
−8.584572813427857880094932529525, −7.46700973501973952641894011275, −6.92378515254965284562990938842, −6.18568649291120742988392314760, −5.13958106514169430388553035937, −4.92919747500958820427916501406, −3.36949581819930994956200965924, −2.68086717743184950517514705121, −1.56960403191792476739650508836, −0.70896788283716652567581280321,
0.984852313685248351949572905090, 2.13495859527148901253286935177, 2.66998497560768930962996298893, 3.82863083860813835195992086317, 4.78257104180039476059286387868, 5.91367244059823164269930426030, 6.17816776862389803231161281185, 7.18858590718970587644935989222, 7.63950483232081335852261809306, 8.535111183990424746610367550914