L(s) = 1 | + (0.951 + 0.309i)3-s − 1.61i·7-s + (−1.30 − 0.951i)13-s + (−0.587 + 0.190i)19-s + (0.500 − 1.53i)21-s + (−0.587 − 0.809i)23-s + (−0.587 − 0.809i)27-s + (−0.190 + 0.587i)29-s + (0.951 − 0.309i)31-s + (−0.809 − 0.587i)37-s + (−0.951 − 1.30i)39-s + 0.618i·43-s + (1.53 + 0.5i)47-s − 1.61·49-s + (0.309 − 0.951i)53-s + ⋯ |
L(s) = 1 | + (0.951 + 0.309i)3-s − 1.61i·7-s + (−1.30 − 0.951i)13-s + (−0.587 + 0.190i)19-s + (0.500 − 1.53i)21-s + (−0.587 − 0.809i)23-s + (−0.587 − 0.809i)27-s + (−0.190 + 0.587i)29-s + (0.951 − 0.309i)31-s + (−0.809 − 0.587i)37-s + (−0.951 − 1.30i)39-s + 0.618i·43-s + (1.53 + 0.5i)47-s − 1.61·49-s + (0.309 − 0.951i)53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0314 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0314 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.349230309\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.349230309\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
good | 3 | \( 1 + (-0.951 - 0.309i)T + (0.809 + 0.587i)T^{2} \) |
| 7 | \( 1 + 1.61iT - T^{2} \) |
| 11 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 13 | \( 1 + (1.30 + 0.951i)T + (0.309 + 0.951i)T^{2} \) |
| 17 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 19 | \( 1 + (0.587 - 0.190i)T + (0.809 - 0.587i)T^{2} \) |
| 23 | \( 1 + (0.587 + 0.809i)T + (-0.309 + 0.951i)T^{2} \) |
| 29 | \( 1 + (0.190 - 0.587i)T + (-0.809 - 0.587i)T^{2} \) |
| 31 | \( 1 + (-0.951 + 0.309i)T + (0.809 - 0.587i)T^{2} \) |
| 37 | \( 1 + (0.809 + 0.587i)T + (0.309 + 0.951i)T^{2} \) |
| 41 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 43 | \( 1 - 0.618iT - T^{2} \) |
| 47 | \( 1 + (-1.53 - 0.5i)T + (0.809 + 0.587i)T^{2} \) |
| 53 | \( 1 + (-0.309 + 0.951i)T + (-0.809 - 0.587i)T^{2} \) |
| 59 | \( 1 + (-0.363 + 0.5i)T + (-0.309 - 0.951i)T^{2} \) |
| 61 | \( 1 + (0.809 - 0.587i)T + (0.309 - 0.951i)T^{2} \) |
| 67 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 71 | \( 1 + (-1.53 - 0.5i)T + (0.809 + 0.587i)T^{2} \) |
| 73 | \( 1 + (-0.809 + 0.587i)T + (0.309 - 0.951i)T^{2} \) |
| 79 | \( 1 + (0.587 + 0.190i)T + (0.809 + 0.587i)T^{2} \) |
| 83 | \( 1 + (-0.951 + 0.309i)T + (0.809 - 0.587i)T^{2} \) |
| 89 | \( 1 + (0.309 - 0.951i)T^{2} \) |
| 97 | \( 1 + (0.5 - 1.53i)T + (-0.809 - 0.587i)T^{2} \) |
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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
−8.241029378551049914888964238678, −7.87253386156096891443956916207, −7.14483210249897194686944817386, −6.40662388691243263686420986219, −5.31796458978335664885111823196, −4.40461626678952350293642083608, −3.84792808343963239516333998619, −2.99886694531422314760539862061, −2.17167303544939574482900598708, −0.61141137689783561563809075641,
1.95519868848024717364140168301, 2.32371117744549810633837965577, 3.10860307334345410814658634908, 4.22706954287113224219582222954, 5.15902263361418326480020731209, 5.80147462905595997649701603856, 6.72845185826677989158284366430, 7.46124575027894671271054810777, 8.253595338696294372357407372546, 8.746491133000433485963573418523