| L(s) = 1 | − 2.42·3-s − 1.20i·5-s − i·7-s + 2.90·9-s − i·11-s + (−2.92 + 2.10i)13-s + 2.91i·15-s − 0.0204·17-s + 1.17i·19-s + 2.42i·21-s − 3.64·23-s + 3.55·25-s + 0.238·27-s + 9.92·29-s − 1.43i·31-s + ⋯ |
| L(s) = 1 | − 1.40·3-s − 0.536i·5-s − 0.377i·7-s + 0.967·9-s − 0.301i·11-s + (−0.812 + 0.583i)13-s + 0.752i·15-s − 0.00496·17-s + 0.269i·19-s + 0.530i·21-s − 0.760·23-s + 0.711·25-s + 0.0458·27-s + 1.84·29-s − 0.257i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.812 + 0.583i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.812 + 0.583i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5154299149\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5154299149\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + iT \) |
| 11 | \( 1 + iT \) |
| 13 | \( 1 + (2.92 - 2.10i)T \) |
| good | 3 | \( 1 + 2.42T + 3T^{2} \) |
| 5 | \( 1 + 1.20iT - 5T^{2} \) |
| 17 | \( 1 + 0.0204T + 17T^{2} \) |
| 19 | \( 1 - 1.17iT - 19T^{2} \) |
| 23 | \( 1 + 3.64T + 23T^{2} \) |
| 29 | \( 1 - 9.92T + 29T^{2} \) |
| 31 | \( 1 + 1.43iT - 31T^{2} \) |
| 37 | \( 1 + 8.76iT - 37T^{2} \) |
| 41 | \( 1 - 4.94iT - 41T^{2} \) |
| 43 | \( 1 - 0.228T + 43T^{2} \) |
| 47 | \( 1 + 9.65iT - 47T^{2} \) |
| 53 | \( 1 + 4.95T + 53T^{2} \) |
| 59 | \( 1 - 7.31iT - 59T^{2} \) |
| 61 | \( 1 - 1.17T + 61T^{2} \) |
| 67 | \( 1 + 7.58iT - 67T^{2} \) |
| 71 | \( 1 + 11.7iT - 71T^{2} \) |
| 73 | \( 1 - 11.4iT - 73T^{2} \) |
| 79 | \( 1 - 10.1T + 79T^{2} \) |
| 83 | \( 1 - 3.97iT - 83T^{2} \) |
| 89 | \( 1 - 0.313iT - 89T^{2} \) |
| 97 | \( 1 - 6.85iT - 97T^{2} \) |
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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
−8.119733966889673461777116004334, −7.25394809398236978067983488966, −6.53416736764089884353763322991, −5.97631030251048460672040232678, −5.04389785380936632633137479341, −4.68024430356854680980504986395, −3.75824771210099845924811030524, −2.43828244386986724013366649223, −1.17808548792445491602030589651, −0.23398048077906223682456190371,
1.02555948147868304470195724022, 2.42520150664463332401337438259, 3.21931170414152926433073258348, 4.63306515923897515325651981913, 4.93549457144174742063875973784, 5.87339890791652108513010682925, 6.46738407125085666775197571570, 7.02506188683779518664924995820, 7.906277791848966462612726908110, 8.679391272413747333681697817572
