L(s) = 1 | − 3.20i·3-s − 1.49i·5-s + 2.89i·7-s − 7.25·9-s + 0.590i·11-s + 6.18·13-s − 4.80·15-s + (3.80 − 1.59i)17-s − 2.41·19-s + 9.27·21-s − 6.41i·23-s + 2.75·25-s + 13.6i·27-s − 6.45i·29-s + 1.17i·31-s + ⋯ |
L(s) = 1 | − 1.84i·3-s − 0.670i·5-s + 1.09i·7-s − 2.41·9-s + 0.177i·11-s + 1.71·13-s − 1.23·15-s + (0.922 − 0.387i)17-s − 0.554·19-s + 2.02·21-s − 1.33i·23-s + 0.550·25-s + 2.62i·27-s − 1.19i·29-s + 0.210i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.922 + 0.387i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.922 + 0.387i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.822066267\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.822066267\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 17 | \( 1 + (-3.80 + 1.59i)T \) |
| 59 | \( 1 - T \) |
good | 3 | \( 1 + 3.20iT - 3T^{2} \) |
| 5 | \( 1 + 1.49iT - 5T^{2} \) |
| 7 | \( 1 - 2.89iT - 7T^{2} \) |
| 11 | \( 1 - 0.590iT - 11T^{2} \) |
| 13 | \( 1 - 6.18T + 13T^{2} \) |
| 19 | \( 1 + 2.41T + 19T^{2} \) |
| 23 | \( 1 + 6.41iT - 23T^{2} \) |
| 29 | \( 1 + 6.45iT - 29T^{2} \) |
| 31 | \( 1 - 1.17iT - 31T^{2} \) |
| 37 | \( 1 - 0.792iT - 37T^{2} \) |
| 41 | \( 1 + 3.09iT - 41T^{2} \) |
| 43 | \( 1 + 0.984T + 43T^{2} \) |
| 47 | \( 1 - 11.2T + 47T^{2} \) |
| 53 | \( 1 + 4.09T + 53T^{2} \) |
| 61 | \( 1 - 8.04iT - 61T^{2} \) |
| 67 | \( 1 + 13.0T + 67T^{2} \) |
| 71 | \( 1 + 7.20iT - 71T^{2} \) |
| 73 | \( 1 - 3.29iT - 73T^{2} \) |
| 79 | \( 1 + 10.9iT - 79T^{2} \) |
| 83 | \( 1 - 1.87T + 83T^{2} \) |
| 89 | \( 1 + 2.40T + 89T^{2} \) |
| 97 | \( 1 + 11.0iT - 97T^{2} \) |
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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
−8.168767495369541298930964686892, −7.46193615365891406248382110406, −6.56553110815117313596511363588, −5.97779684282057221254416512889, −5.56465293251587745357957717454, −4.40254765242873356957465209863, −3.09874483502615663461284949497, −2.32747809326274108049382529979, −1.44231828950491129875372243654, −0.59737689824590816945473272524,
1.20226394680276799487553493956, 2.93576095885336349444624563257, 3.73597765907413395056413260485, 3.82686670715728144335934259348, 4.91790470582103613868031723947, 5.71887101403394264987909797617, 6.36350587138327826473602413252, 7.36101565660134464773847401750, 8.219229493068584089560466434149, 8.908987088239041660009962503905