L(s) = 1 | + (0.173 − 0.984i)3-s + (−0.766 + 0.642i)4-s + (−0.939 − 0.342i)9-s + (0.5 + 0.866i)12-s + (0.326 + 1.85i)13-s + (0.173 − 0.984i)16-s + (−0.5 + 0.866i)19-s + (−0.173 − 0.984i)25-s + (−0.5 + 0.866i)27-s + (0.766 + 1.32i)31-s + (0.939 − 0.342i)36-s + 1.96i·37-s + 1.87·39-s + (−0.266 − 0.223i)43-s + (−0.939 − 0.342i)48-s + ⋯ |
L(s) = 1 | + (0.173 − 0.984i)3-s + (−0.766 + 0.642i)4-s + (−0.939 − 0.342i)9-s + (0.5 + 0.866i)12-s + (0.326 + 1.85i)13-s + (0.173 − 0.984i)16-s + (−0.5 + 0.866i)19-s + (−0.173 − 0.984i)25-s + (−0.5 + 0.866i)27-s + (0.766 + 1.32i)31-s + (0.939 − 0.342i)36-s + 1.96i·37-s + 1.87·39-s + (−0.266 − 0.223i)43-s + (−0.939 − 0.342i)48-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2793 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.671 - 0.740i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2793 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.671 - 0.740i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8756466445\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8756466445\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.173 + 0.984i)T \) |
| 7 | \( 1 \) |
| 19 | \( 1 + (0.5 - 0.866i)T \) |
good | 2 | \( 1 + (0.766 - 0.642i)T^{2} \) |
| 5 | \( 1 + (0.173 + 0.984i)T^{2} \) |
| 11 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 + (-0.326 - 1.85i)T + (-0.939 + 0.342i)T^{2} \) |
| 17 | \( 1 + (0.766 - 0.642i)T^{2} \) |
| 23 | \( 1 + (-0.173 + 0.984i)T^{2} \) |
| 29 | \( 1 + (0.766 + 0.642i)T^{2} \) |
| 31 | \( 1 + (-0.766 - 1.32i)T + (-0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 - 1.96iT - T^{2} \) |
| 41 | \( 1 + (0.939 + 0.342i)T^{2} \) |
| 43 | \( 1 + (0.266 + 0.223i)T + (0.173 + 0.984i)T^{2} \) |
| 47 | \( 1 + (0.766 + 0.642i)T^{2} \) |
| 53 | \( 1 + (0.173 - 0.984i)T^{2} \) |
| 59 | \( 1 + (-0.766 + 0.642i)T^{2} \) |
| 61 | \( 1 + (-0.826 - 0.984i)T + (-0.173 + 0.984i)T^{2} \) |
| 67 | \( 1 + (-0.233 + 0.642i)T + (-0.766 - 0.642i)T^{2} \) |
| 71 | \( 1 + (0.173 + 0.984i)T^{2} \) |
| 73 | \( 1 + (-0.673 - 0.118i)T + (0.939 + 0.342i)T^{2} \) |
| 79 | \( 1 + (-1.26 - 0.223i)T + (0.939 + 0.342i)T^{2} \) |
| 83 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 89 | \( 1 + (0.939 - 0.342i)T^{2} \) |
| 97 | \( 1 + (-0.939 + 0.342i)T + (0.766 - 0.642i)T^{2} \) |
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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
−8.789108078231855338346435219014, −8.395433708400424953023200059000, −7.72211384550965562831810656361, −6.67309757852154338332769288653, −6.42560153442304587196720201755, −5.13803011437355749258103386355, −4.28634591160026558106083749072, −3.48613729844348860055313343403, −2.42371076888873624515443201298, −1.35074426434797745995099839904,
0.59781308088336191949865761881, 2.36170627648345842047261750464, 3.44040312021017779341716224720, 4.14598495543983918908964801258, 5.09620598327130620208992632977, 5.54193307198223537649088243892, 6.30179447144839747937629523180, 7.66258762209651927777942920191, 8.295290488430041898843808910311, 9.034820468552344851215424127829