| L(s) = 1 | + 7·3-s − 79·5-s − 50·7-s − 194·9-s + 121·11-s − 380·13-s − 553·15-s − 1.15e3·17-s − 1.82e3·19-s − 350·21-s + 3.59e3·23-s + 3.11e3·25-s − 3.05e3·27-s + 8.03e3·29-s − 2.94e3·31-s + 847·33-s + 3.95e3·35-s + 6.97e3·37-s − 2.66e3·39-s − 520·41-s − 2.48e3·43-s + 1.53e4·45-s − 6.92e3·47-s − 1.43e4·49-s − 8.07e3·51-s − 1.37e4·53-s − 9.55e3·55-s + ⋯ |
| L(s) = 1 | + 0.449·3-s − 1.41·5-s − 0.385·7-s − 0.798·9-s + 0.301·11-s − 0.623·13-s − 0.634·15-s − 0.968·17-s − 1.15·19-s − 0.173·21-s + 1.41·23-s + 0.997·25-s − 0.807·27-s + 1.77·29-s − 0.550·31-s + 0.135·33-s + 0.545·35-s + 0.838·37-s − 0.280·39-s − 0.0483·41-s − 0.205·43-s + 1.12·45-s − 0.456·47-s − 0.851·49-s − 0.434·51-s − 0.670·53-s − 0.426·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 44 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 44 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 - p^{2} T \) |
| good | 3 | \( 1 - 7 T + p^{5} T^{2} \) |
| 5 | \( 1 + 79 T + p^{5} T^{2} \) |
| 7 | \( 1 + 50 T + p^{5} T^{2} \) |
| 13 | \( 1 + 380 T + p^{5} T^{2} \) |
| 17 | \( 1 + 1154 T + p^{5} T^{2} \) |
| 19 | \( 1 + 96 p T + p^{5} T^{2} \) |
| 23 | \( 1 - 3591 T + p^{5} T^{2} \) |
| 29 | \( 1 - 8032 T + p^{5} T^{2} \) |
| 31 | \( 1 + 95 p T + p^{5} T^{2} \) |
| 37 | \( 1 - 6979 T + p^{5} T^{2} \) |
| 41 | \( 1 + 520 T + p^{5} T^{2} \) |
| 43 | \( 1 + 2486 T + p^{5} T^{2} \) |
| 47 | \( 1 + 6920 T + p^{5} T^{2} \) |
| 53 | \( 1 + 13718 T + p^{5} T^{2} \) |
| 59 | \( 1 + 31779 T + p^{5} T^{2} \) |
| 61 | \( 1 - 34156 T + p^{5} T^{2} \) |
| 67 | \( 1 + 61503 T + p^{5} T^{2} \) |
| 71 | \( 1 + 14971 T + p^{5} T^{2} \) |
| 73 | \( 1 + 36444 T + p^{5} T^{2} \) |
| 79 | \( 1 + 28538 T + p^{5} T^{2} \) |
| 83 | \( 1 - 77482 T + p^{5} T^{2} \) |
| 89 | \( 1 - 36271 T + p^{5} T^{2} \) |
| 97 | \( 1 + 49799 T + p^{5} T^{2} \) |
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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
−14.57919188660308801254046083493, −13.06782038932331330030150199669, −11.88846666005788328877833143839, −10.87879525470777677183044116768, −9.067797044210769344913095584551, −8.083409346927810532746450243894, −6.68942492282015352613928744340, −4.48674472090030800382158243308, −2.96268618360357177696162073441, 0,
2.96268618360357177696162073441, 4.48674472090030800382158243308, 6.68942492282015352613928744340, 8.083409346927810532746450243894, 9.067797044210769344913095584551, 10.87879525470777677183044116768, 11.88846666005788328877833143839, 13.06782038932331330030150199669, 14.57919188660308801254046083493
