L(s) = 1 | + (0.770 + 0.637i)5-s + (−0.982 − 0.187i)9-s + (0.245 + 0.360i)13-s + (1.42 + 0.842i)17-s + (0.187 + 0.982i)25-s + (−0.961 + 0.0604i)29-s + (0.0175 + 0.0603i)37-s + (0.362 − 0.340i)41-s + (−0.637 − 0.770i)45-s + (0.587 + 0.809i)49-s + (0.288 + 1.29i)53-s + (1.12 + 1.05i)61-s + (−0.0410 + 0.434i)65-s + (−0.621 − 1.72i)73-s + (0.929 + 0.368i)81-s + ⋯ |
L(s) = 1 | + (0.770 + 0.637i)5-s + (−0.982 − 0.187i)9-s + (0.245 + 0.360i)13-s + (1.42 + 0.842i)17-s + (0.187 + 0.982i)25-s + (−0.961 + 0.0604i)29-s + (0.0175 + 0.0603i)37-s + (0.362 − 0.340i)41-s + (−0.637 − 0.770i)45-s + (0.587 + 0.809i)49-s + (0.288 + 1.29i)53-s + (1.12 + 1.05i)61-s + (−0.0410 + 0.434i)65-s + (−0.621 − 1.72i)73-s + (0.929 + 0.368i)81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.582 - 0.812i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.582 - 0.812i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.407100426\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.407100426\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-0.770 - 0.637i)T \) |
good | 3 | \( 1 + (0.982 + 0.187i)T^{2} \) |
| 7 | \( 1 + (-0.587 - 0.809i)T^{2} \) |
| 11 | \( 1 + (0.968 + 0.248i)T^{2} \) |
| 13 | \( 1 + (-0.245 - 0.360i)T + (-0.368 + 0.929i)T^{2} \) |
| 17 | \( 1 + (-1.42 - 0.842i)T + (0.481 + 0.876i)T^{2} \) |
| 19 | \( 1 + (0.187 + 0.982i)T^{2} \) |
| 23 | \( 1 + (0.998 - 0.0627i)T^{2} \) |
| 29 | \( 1 + (0.961 - 0.0604i)T + (0.992 - 0.125i)T^{2} \) |
| 31 | \( 1 + (0.876 - 0.481i)T^{2} \) |
| 37 | \( 1 + (-0.0175 - 0.0603i)T + (-0.844 + 0.535i)T^{2} \) |
| 41 | \( 1 + (-0.362 + 0.340i)T + (0.0627 - 0.998i)T^{2} \) |
| 43 | \( 1 + (-0.951 - 0.309i)T^{2} \) |
| 47 | \( 1 + (0.684 + 0.728i)T^{2} \) |
| 53 | \( 1 + (-0.288 - 1.29i)T + (-0.904 + 0.425i)T^{2} \) |
| 59 | \( 1 + (0.637 - 0.770i)T^{2} \) |
| 61 | \( 1 + (-1.12 - 1.05i)T + (0.0627 + 0.998i)T^{2} \) |
| 67 | \( 1 + (0.125 - 0.992i)T^{2} \) |
| 71 | \( 1 + (0.728 - 0.684i)T^{2} \) |
| 73 | \( 1 + (0.621 + 1.72i)T + (-0.770 + 0.637i)T^{2} \) |
| 79 | \( 1 + (0.187 - 0.982i)T^{2} \) |
| 83 | \( 1 + (-0.982 + 0.187i)T^{2} \) |
| 89 | \( 1 + (-0.226 - 0.106i)T + (0.637 + 0.770i)T^{2} \) |
| 97 | \( 1 + (-1.24 + 1.41i)T + (-0.125 - 0.992i)T^{2} \) |
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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
−8.916423807373751093241132819093, −7.927690483203416434619939715818, −7.32446077147451859862841431807, −6.33089047225590656743176989859, −5.84811195801402687008396596896, −5.29100431967538723515627111299, −3.99713553691727570367242990152, −3.24401606012615181768652259894, −2.42864684212833163779930150377, −1.36719419688712354284960110862,
0.846091326562409724932860429908, 2.07475412056706664390752638820, 2.96441586697343057749602187735, 3.85713005085998076902181338488, 5.12532529734827911874176045139, 5.41875155405395749456480423767, 6.10665747514458259897609879781, 7.06695682920848949207589300151, 7.968780688664796667810087495632, 8.474856784617029059999684466129