| L(s) = 1 | + 2·3-s − 7-s + 9-s − 2·11-s − 4·13-s + 6·17-s − 2·21-s + 23-s − 4·27-s + 2·29-s − 4·31-s − 4·33-s − 8·39-s + 6·41-s − 6·43-s + 49-s + 12·51-s − 12·53-s − 10·59-s − 2·61-s − 63-s + 2·67-s + 2·69-s − 8·71-s − 2·73-s + 2·77-s − 8·79-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.377·7-s + 1/3·9-s − 0.603·11-s − 1.10·13-s + 1.45·17-s − 0.436·21-s + 0.208·23-s − 0.769·27-s + 0.371·29-s − 0.718·31-s − 0.696·33-s − 1.28·39-s + 0.937·41-s − 0.914·43-s + 1/7·49-s + 1.68·51-s − 1.64·53-s − 1.30·59-s − 0.256·61-s − 0.125·63-s + 0.244·67-s + 0.240·69-s − 0.949·71-s − 0.234·73-s + 0.227·77-s − 0.900·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 257600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 257600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−12.90223655035630, −12.79269204013694, −12.21581496178246, −11.80595459342513, −11.17245326649331, −10.66843241908531, −9.993686748375115, −9.901830305474318, −9.289431093003888, −8.946384142486934, −8.354233181522469, −7.840137113640454, −7.507363164380555, −7.255147891976159, −6.397887756641366, −5.914625867103023, −5.408500903211137, −4.779506810941821, −4.421146630546089, −3.423225245145056, −3.287231145882762, −2.836237246769491, −2.143464586787462, −1.712047934668081, −0.7757331602921673, 0,
0.7757331602921673, 1.712047934668081, 2.143464586787462, 2.836237246769491, 3.287231145882762, 3.423225245145056, 4.421146630546089, 4.779506810941821, 5.408500903211137, 5.914625867103023, 6.397887756641366, 7.255147891976159, 7.507363164380555, 7.840137113640454, 8.354233181522469, 8.946384142486934, 9.289431093003888, 9.901830305474318, 9.993686748375115, 10.66843241908531, 11.17245326649331, 11.80595459342513, 12.21581496178246, 12.79269204013694, 12.90223655035630
