L(s) = 1 | + (2.94 + 0.581i)3-s + 11.4i·7-s + (8.32 + 3.42i)9-s + 8.48i·11-s − 10i·13-s + 3.55·17-s − 10.9·19-s + (−6.67 + 33.8i)21-s + 17.6·23-s + (22.5 + 14.9i)27-s + 26.8i·29-s − 8·31-s + (−4.93 + 24.9i)33-s + 59.9i·37-s + (5.81 − 29.4i)39-s + ⋯ |
L(s) = 1 | + (0.981 + 0.193i)3-s + 1.64i·7-s + (0.924 + 0.380i)9-s + 0.771i·11-s − 0.769i·13-s + 0.209·17-s − 0.577·19-s + (−0.317 + 1.60i)21-s + 0.767·23-s + (0.833 + 0.552i)27-s + 0.925i·29-s − 0.258·31-s + (−0.149 + 0.756i)33-s + 1.62i·37-s + (0.149 − 0.754i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.265 - 0.964i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.265 - 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(2.559417928\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.559417928\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-2.94 - 0.581i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - 11.4iT - 49T^{2} \) |
| 11 | \( 1 - 8.48iT - 121T^{2} \) |
| 13 | \( 1 + 10iT - 169T^{2} \) |
| 17 | \( 1 - 3.55T + 289T^{2} \) |
| 19 | \( 1 + 10.9T + 361T^{2} \) |
| 23 | \( 1 - 17.6T + 529T^{2} \) |
| 29 | \( 1 - 26.8iT - 841T^{2} \) |
| 31 | \( 1 + 8T + 961T^{2} \) |
| 37 | \( 1 - 59.9iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 20.5iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 42.4iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 88.2T + 2.20e3T^{2} \) |
| 53 | \( 1 + 3.55T + 2.80e3T^{2} \) |
| 59 | \( 1 - 77.7iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 21.9T + 3.72e3T^{2} \) |
| 67 | \( 1 + 53.5iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 69.2iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 12.0iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 9.02T + 6.24e3T^{2} \) |
| 83 | \( 1 - 0.688T + 6.88e3T^{2} \) |
| 89 | \( 1 - 7.10iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 111. iT - 9.40e3T^{2} \) |
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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
−9.698069767668252306090998603239, −8.799943028406997461891580744314, −8.462680928470050188179622717248, −7.47703582059622538616557930625, −6.55185530925664557019900860949, −5.40052076559708602294061311604, −4.72777018567641193860910351077, −3.37912556069932863141239464809, −2.62295010198560608129286324225, −1.67051067142866651265701900549,
0.65435201397561195939185138157, 1.81566531623860650246338519208, 3.17704077552027946194930910494, 3.94109316726922593877110429975, 4.69911941429307078562941517786, 6.25723150841403629989909868149, 6.98218159158209886141304809198, 7.71570400330137303338895517436, 8.402194796357986587150968425727, 9.341304145894934014727557869104