L(s) = 1 | + i·3-s + 2i·7-s + 2·9-s − 3·11-s + 4i·13-s + 3i·17-s − 5·19-s − 2·21-s − 6i·23-s + 5i·27-s − 2·31-s − 3i·33-s + 2i·37-s − 4·39-s − 3·41-s + ⋯ |
L(s) = 1 | + 0.577i·3-s + 0.755i·7-s + 0.666·9-s − 0.904·11-s + 1.10i·13-s + 0.727i·17-s − 1.14·19-s − 0.436·21-s − 1.25i·23-s + 0.962i·27-s − 0.359·31-s − 0.522i·33-s + 0.328i·37-s − 0.640·39-s − 0.468·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.069747472\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.069747472\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
good | 3 | \( 1 - iT - 3T^{2} \) |
| 7 | \( 1 - 2iT - 7T^{2} \) |
| 11 | \( 1 + 3T + 11T^{2} \) |
| 13 | \( 1 - 4iT - 13T^{2} \) |
| 17 | \( 1 - 3iT - 17T^{2} \) |
| 19 | \( 1 + 5T + 19T^{2} \) |
| 23 | \( 1 + 6iT - 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + 2T + 31T^{2} \) |
| 37 | \( 1 - 2iT - 37T^{2} \) |
| 41 | \( 1 + 3T + 41T^{2} \) |
| 43 | \( 1 + 4iT - 43T^{2} \) |
| 47 | \( 1 - 12iT - 47T^{2} \) |
| 53 | \( 1 + 6iT - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 - 13iT - 67T^{2} \) |
| 71 | \( 1 + 12T + 71T^{2} \) |
| 73 | \( 1 - 11iT - 73T^{2} \) |
| 79 | \( 1 + 10T + 79T^{2} \) |
| 83 | \( 1 + 9iT - 83T^{2} \) |
| 89 | \( 1 + 15T + 89T^{2} \) |
| 97 | \( 1 + 2iT - 97T^{2} \) |
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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
−9.833473105060152715516327423133, −8.839874986700037821240829008085, −8.454802171153744946770242249215, −7.30643623631575389702948724994, −6.48520933267338993531911959906, −5.63412177363564889887943413343, −4.60083420206572865839383475998, −4.08701952912088618592638500784, −2.73991896352042329987197508111, −1.77825955755040677392653954616,
0.39734728218883045740931514747, 1.71065455482548057711580589783, 2.90176831911550508098756468628, 3.96626328181030401313763663332, 4.95642289359197992827615876786, 5.81944050046625790123303105400, 6.85912414369834048784166163008, 7.52265525667723835458661791976, 7.979946653368058210238291303002, 9.033129279835063836287112328697