| L(s) = 1 | − 3-s + 5-s − 2·9-s − 2·11-s − 15-s + 4·17-s + 2·19-s + 23-s + 25-s + 5·27-s − 9·29-s + 4·31-s + 2·33-s − 4·37-s + 41-s − 9·43-s − 2·45-s − 4·51-s + 10·53-s − 2·55-s − 2·57-s + 10·59-s − 9·61-s − 5·67-s − 69-s + 14·71-s + 12·73-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s − 2/3·9-s − 0.603·11-s − 0.258·15-s + 0.970·17-s + 0.458·19-s + 0.208·23-s + 1/5·25-s + 0.962·27-s − 1.67·29-s + 0.718·31-s + 0.348·33-s − 0.657·37-s + 0.156·41-s − 1.37·43-s − 0.298·45-s − 0.560·51-s + 1.37·53-s − 0.269·55-s − 0.264·57-s + 1.30·59-s − 1.15·61-s − 0.610·67-s − 0.120·69-s + 1.66·71-s + 1.40·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.380926189\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.380926189\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - T + p T^{2} \) |
| 43 | \( 1 + 9 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 + 9 T + p T^{2} \) |
| 67 | \( 1 + 5 T + p T^{2} \) |
| 71 | \( 1 - 14 T + p T^{2} \) |
| 73 | \( 1 - 12 T + p T^{2} \) |
| 79 | \( 1 - 14 T + p T^{2} \) |
| 83 | \( 1 + 11 T + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−16.20109858915776, −15.30967378586983, −14.99967250567562, −14.27982142030783, −13.70811614439200, −13.32886670065591, −12.50530758567064, −12.11383814220833, −11.48153887561283, −10.92223921094365, −10.43436259946736, −9.714877895520411, −9.339613484921433, −8.359268055170991, −8.085992222988294, −7.146936128663758, −6.680692004924267, −5.741159260699111, −5.470085979876711, −4.981862601711040, −3.917638084265173, −3.180012462605698, −2.485951295568571, −1.536507648583156, −0.5307146596073817,
0.5307146596073817, 1.536507648583156, 2.485951295568571, 3.180012462605698, 3.917638084265173, 4.981862601711040, 5.470085979876711, 5.741159260699111, 6.680692004924267, 7.146936128663758, 8.085992222988294, 8.359268055170991, 9.339613484921433, 9.714877895520411, 10.43436259946736, 10.92223921094365, 11.48153887561283, 12.11383814220833, 12.50530758567064, 13.32886670065591, 13.70811614439200, 14.27982142030783, 14.99967250567562, 15.30967378586983, 16.20109858915776
