| L(s) = 1 | − 0.547i·3-s − 10.0i·7-s + 8.69·9-s + 17.2i·11-s − 4.41·13-s + 27.0·17-s − 4.82i·19-s − 5.50·21-s + 15.2i·23-s − 9.69i·27-s + 2.38·29-s − 38.0i·31-s + 9.42·33-s + 16.5·37-s + 2.41i·39-s + ⋯ |
| L(s) = 1 | − 0.182i·3-s − 1.43i·7-s + 0.966·9-s + 1.56i·11-s − 0.339·13-s + 1.58·17-s − 0.254i·19-s − 0.262·21-s + 0.663i·23-s − 0.359i·27-s + 0.0821·29-s − 1.22i·31-s + 0.285·33-s + 0.447·37-s + 0.0619i·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.042169911\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.042169911\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + 0.547iT - 9T^{2} \) |
| 7 | \( 1 + 10.0iT - 49T^{2} \) |
| 11 | \( 1 - 17.2iT - 121T^{2} \) |
| 13 | \( 1 + 4.41T + 169T^{2} \) |
| 17 | \( 1 - 27.0T + 289T^{2} \) |
| 19 | \( 1 + 4.82iT - 361T^{2} \) |
| 23 | \( 1 - 15.2iT - 529T^{2} \) |
| 29 | \( 1 - 2.38T + 841T^{2} \) |
| 31 | \( 1 + 38.0iT - 961T^{2} \) |
| 37 | \( 1 - 16.5T + 1.36e3T^{2} \) |
| 41 | \( 1 + 13.3T + 1.68e3T^{2} \) |
| 43 | \( 1 + 59.7iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 62.4iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 71.5T + 2.80e3T^{2} \) |
| 59 | \( 1 - 68.8iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 40.9T + 3.72e3T^{2} \) |
| 67 | \( 1 + 51.0iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 40.4iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 35.8T + 5.32e3T^{2} \) |
| 79 | \( 1 + 126. iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 75.1iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 106.T + 7.92e3T^{2} \) |
| 97 | \( 1 - 85.4T + 9.40e3T^{2} \) |
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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
−10.10694802586685445454659800022, −9.419740158507746440988333712876, −7.87811394414996331699277842641, −7.32592527737279375774730352920, −6.87491544139087036543417042305, −5.40604015209227143152811764740, −4.38292637126108918997657317823, −3.70139867405348065444736044668, −2.02044033914813353952758879292, −0.860428169207505225838134986380,
1.15606476895195640582383556027, 2.67690308183827298923228200049, 3.58001929051004237569427247163, 4.96116606023835606884205631602, 5.71792545289936610000869571763, 6.52693547048828184603606130912, 7.80388301092248968672210491656, 8.487314639306630708059375965829, 9.331931445392762518003967807595, 10.10024321363636175136099051655
