L(s) = 1 | + (−0.929 + 0.368i)2-s + (0.728 − 0.684i)4-s + (0.992 − 0.125i)5-s + (−0.425 + 0.904i)8-s + (−0.187 − 0.982i)9-s + (−0.876 + 0.481i)10-s + (−0.383 + 1.49i)13-s + (0.0627 − 0.998i)16-s + (1.11 − 0.363i)17-s + (0.535 + 0.844i)18-s + (0.637 − 0.770i)20-s + (0.968 − 0.248i)25-s + (−0.193 − 1.52i)26-s + (−0.473 + 1.84i)29-s + (0.309 + 0.951i)32-s + ⋯ |
L(s) = 1 | + (−0.929 + 0.368i)2-s + (0.728 − 0.684i)4-s + (0.992 − 0.125i)5-s + (−0.425 + 0.904i)8-s + (−0.187 − 0.982i)9-s + (−0.876 + 0.481i)10-s + (−0.383 + 1.49i)13-s + (0.0627 − 0.998i)16-s + (1.11 − 0.363i)17-s + (0.535 + 0.844i)18-s + (0.637 − 0.770i)20-s + (0.968 − 0.248i)25-s + (−0.193 − 1.52i)26-s + (−0.473 + 1.84i)29-s + (0.309 + 0.951i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.981 - 0.192i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.981 - 0.192i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9326924584\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9326924584\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.929 - 0.368i)T \) |
| 5 | \( 1 + (-0.992 + 0.125i)T \) |
| 101 | \( 1 + (0.728 - 0.684i)T \) |
good | 3 | \( 1 + (0.187 + 0.982i)T^{2} \) |
| 7 | \( 1 + (-0.876 + 0.481i)T^{2} \) |
| 11 | \( 1 + (-0.929 + 0.368i)T^{2} \) |
| 13 | \( 1 + (0.383 - 1.49i)T + (-0.876 - 0.481i)T^{2} \) |
| 17 | \( 1 + (-1.11 + 0.363i)T + (0.809 - 0.587i)T^{2} \) |
| 19 | \( 1 + (0.992 - 0.125i)T^{2} \) |
| 23 | \( 1 + (-0.425 - 0.904i)T^{2} \) |
| 29 | \( 1 + (0.473 - 1.84i)T + (-0.876 - 0.481i)T^{2} \) |
| 31 | \( 1 + (-0.876 + 0.481i)T^{2} \) |
| 37 | \( 1 + (-1.05 + 0.872i)T + (0.187 - 0.982i)T^{2} \) |
| 41 | \( 1 + (-1.89 - 0.616i)T + (0.809 + 0.587i)T^{2} \) |
| 43 | \( 1 + (0.968 - 0.248i)T^{2} \) |
| 47 | \( 1 + (0.968 + 0.248i)T^{2} \) |
| 53 | \( 1 + (1.41 + 1.32i)T + (0.0627 + 0.998i)T^{2} \) |
| 59 | \( 1 + (-0.992 - 0.125i)T^{2} \) |
| 61 | \( 1 + (1.23 + 1.31i)T + (-0.0627 + 0.998i)T^{2} \) |
| 67 | \( 1 + (0.187 - 0.982i)T^{2} \) |
| 71 | \( 1 + (0.187 + 0.982i)T^{2} \) |
| 73 | \( 1 + (-1.06 + 1.67i)T + (-0.425 - 0.904i)T^{2} \) |
| 79 | \( 1 + (0.425 - 0.904i)T^{2} \) |
| 83 | \( 1 + (0.425 - 0.904i)T^{2} \) |
| 89 | \( 1 + (1.96 - 0.123i)T + (0.992 - 0.125i)T^{2} \) |
| 97 | \( 1 + (-0.340 - 0.362i)T + (-0.0627 + 0.998i)T^{2} \) |
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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
−9.442292295728758341551194884368, −8.890241779233463990875075382152, −7.79655603448225824935903717533, −6.97713152126458280420836411639, −6.35142458025682252040143082654, −5.65338222354881848540672827031, −4.74884252773858598961110776702, −3.31337391369119573534913250093, −2.18254479807444916078630581915, −1.17312873662249339923462537958,
1.17935478498998393060341010913, 2.44793870237132947073880434385, 2.93448553436740045345949952668, 4.35809621848555892278495325860, 5.74529701146728818017702511684, 5.91528420860578753149835374479, 7.36538360790726004874729454418, 7.81606028198640058803308173627, 8.495757911913502212581778999585, 9.645786089308207248890615212714