L(s) = 1 | + (−0.809 − 0.587i)2-s + (0.309 + 0.951i)4-s + (−0.309 − 0.951i)5-s + (0.309 − 0.951i)8-s + (0.309 − 0.951i)9-s + (−0.309 + 0.951i)10-s + (0.690 − 0.951i)13-s + (−0.809 + 0.587i)16-s + (−0.809 + 0.587i)18-s + (0.809 − 0.587i)20-s + (−0.809 + 0.587i)25-s + (−1.11 + 0.363i)26-s + 32-s + 36-s + (−1.11 + 1.53i)37-s + ⋯ |
L(s) = 1 | + (−0.809 − 0.587i)2-s + (0.309 + 0.951i)4-s + (−0.309 − 0.951i)5-s + (0.309 − 0.951i)8-s + (0.309 − 0.951i)9-s + (−0.309 + 0.951i)10-s + (0.690 − 0.951i)13-s + (−0.809 + 0.587i)16-s + (−0.809 + 0.587i)18-s + (0.809 − 0.587i)20-s + (−0.809 + 0.587i)25-s + (−1.11 + 0.363i)26-s + 32-s + 36-s + (−1.11 + 1.53i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.645 + 0.763i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.645 + 0.763i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6919217471\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6919217471\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.809 + 0.587i)T \) |
| 5 | \( 1 + (0.309 + 0.951i)T \) |
| 101 | \( 1 + (0.309 + 0.951i)T \) |
good | 3 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 7 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 11 | \( 1 + (-0.809 - 0.587i)T^{2} \) |
| 13 | \( 1 + (-0.690 + 0.951i)T + (-0.309 - 0.951i)T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 23 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 29 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 31 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 37 | \( 1 + (1.11 - 1.53i)T + (-0.309 - 0.951i)T^{2} \) |
| 41 | \( 1 + 1.17iT - T^{2} \) |
| 43 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 47 | \( 1 + (-0.809 - 0.587i)T^{2} \) |
| 53 | \( 1 + (-0.5 + 1.53i)T + (-0.809 - 0.587i)T^{2} \) |
| 59 | \( 1 + (0.309 - 0.951i)T^{2} \) |
| 61 | \( 1 + (1.80 - 0.587i)T + (0.809 - 0.587i)T^{2} \) |
| 67 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 71 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 73 | \( 1 + (-0.5 - 0.363i)T + (0.309 + 0.951i)T^{2} \) |
| 79 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 83 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 89 | \( 1 + (-1.11 + 1.53i)T + (-0.309 - 0.951i)T^{2} \) |
| 97 | \( 1 + (-1.11 + 0.363i)T + (0.809 - 0.587i)T^{2} \) |
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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
−8.838532800413316186356229126430, −8.623942247803537359734441233730, −7.73373430172825447296468840871, −6.92187553879443689230532831758, −5.96066621176332774393976822401, −4.87461076008752514671022557476, −3.81784016151141779927513482991, −3.23765945145332478391620345890, −1.72693449784970872343345275743, −0.68170303082246951370126154162,
1.62426326113691110490032879865, 2.59960452678633028799247511591, 3.94055897692908527124064450916, 4.90622418532061322436663205004, 5.98279915397511520794653465253, 6.60137555461894075205745971341, 7.44493332677726858175057821227, 7.84676668693807865504130123939, 8.844006895617936948943273797466, 9.468551175029996591695518720783