L(s) = 1 | − 2.26·3-s − 5-s − 1.11·7-s + 2.11·9-s − 5.37·11-s + 2.26·15-s + 7.90·19-s + 2.52·21-s + 6.49·23-s + 25-s + 2·27-s + 3.63·29-s − 3.14·31-s + 12.1·33-s + 1.11·35-s − 7.40·37-s − 4.75·41-s + 4.78·43-s − 2.11·45-s + 6.16·47-s − 5.75·49-s + 0.292·53-s + 5.37·55-s − 17.8·57-s + 11.3·59-s − 5.34·61-s − 2.36·63-s + ⋯ |
L(s) = 1 | − 1.30·3-s − 0.447·5-s − 0.421·7-s + 0.705·9-s − 1.62·11-s + 0.583·15-s + 1.81·19-s + 0.550·21-s + 1.35·23-s + 0.200·25-s + 0.384·27-s + 0.675·29-s − 0.565·31-s + 2.11·33-s + 0.188·35-s − 1.21·37-s − 0.742·41-s + 0.729·43-s − 0.315·45-s + 0.898·47-s − 0.822·49-s + 0.0401·53-s + 0.725·55-s − 2.36·57-s + 1.48·59-s − 0.684·61-s − 0.297·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 13 | \( 1 \) |
good | 3 | \( 1 + 2.26T + 3T^{2} \) |
| 7 | \( 1 + 1.11T + 7T^{2} \) |
| 11 | \( 1 + 5.37T + 11T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 - 7.90T + 19T^{2} \) |
| 23 | \( 1 - 6.49T + 23T^{2} \) |
| 29 | \( 1 - 3.63T + 29T^{2} \) |
| 31 | \( 1 + 3.14T + 31T^{2} \) |
| 37 | \( 1 + 7.40T + 37T^{2} \) |
| 41 | \( 1 + 4.75T + 41T^{2} \) |
| 43 | \( 1 - 4.78T + 43T^{2} \) |
| 47 | \( 1 - 6.16T + 47T^{2} \) |
| 53 | \( 1 - 0.292T + 53T^{2} \) |
| 59 | \( 1 - 11.3T + 59T^{2} \) |
| 61 | \( 1 + 5.34T + 61T^{2} \) |
| 67 | \( 1 - 11.8T + 67T^{2} \) |
| 71 | \( 1 - 3.43T + 71T^{2} \) |
| 73 | \( 1 + 4.59T + 73T^{2} \) |
| 79 | \( 1 + 10.8T + 79T^{2} \) |
| 83 | \( 1 + 7.40T + 83T^{2} \) |
| 89 | \( 1 + 14.8T + 89T^{2} \) |
| 97 | \( 1 + 3.76T + 97T^{2} \) |
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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
−8.146536950362588763111214233918, −7.23623588741606551383490496820, −6.89604110434204884832650618622, −5.70629146729204416269836977275, −5.31508567230077005618969981159, −4.71545020971716465221253076056, −3.42775434502845963608355215715, −2.70539153527588991816632057526, −1.04606007879802810539920122569, 0,
1.04606007879802810539920122569, 2.70539153527588991816632057526, 3.42775434502845963608355215715, 4.71545020971716465221253076056, 5.31508567230077005618969981159, 5.70629146729204416269836977275, 6.89604110434204884832650618622, 7.23623588741606551383490496820, 8.146536950362588763111214233918