L(s) = 1 | + 3-s − 5-s + 7-s + 9-s − 11-s − 3.04·13-s − 15-s + 1.69·17-s − 3.69·19-s + 21-s + 2.40·23-s + 25-s + 27-s − 7.25·29-s + 3.76·31-s − 33-s − 35-s − 0.952·37-s − 3.04·39-s − 4.84·41-s + 11.7·43-s − 45-s + 8.03·47-s + 49-s + 1.69·51-s + 4.67·53-s + 55-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.447·5-s + 0.377·7-s + 0.333·9-s − 0.301·11-s − 0.845·13-s − 0.258·15-s + 0.409·17-s − 0.846·19-s + 0.218·21-s + 0.501·23-s + 0.200·25-s + 0.192·27-s − 1.34·29-s + 0.675·31-s − 0.174·33-s − 0.169·35-s − 0.156·37-s − 0.487·39-s − 0.757·41-s + 1.78·43-s − 0.149·45-s + 1.17·47-s + 0.142·49-s + 0.236·51-s + 0.642·53-s + 0.134·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
good | 13 | \( 1 + 3.04T + 13T^{2} \) |
| 17 | \( 1 - 1.69T + 17T^{2} \) |
| 19 | \( 1 + 3.69T + 19T^{2} \) |
| 23 | \( 1 - 2.40T + 23T^{2} \) |
| 29 | \( 1 + 7.25T + 29T^{2} \) |
| 31 | \( 1 - 3.76T + 31T^{2} \) |
| 37 | \( 1 + 0.952T + 37T^{2} \) |
| 41 | \( 1 + 4.84T + 41T^{2} \) |
| 43 | \( 1 - 11.7T + 43T^{2} \) |
| 47 | \( 1 - 8.03T + 47T^{2} \) |
| 53 | \( 1 - 4.67T + 53T^{2} \) |
| 59 | \( 1 + 10.5T + 59T^{2} \) |
| 61 | \( 1 + 4.40T + 61T^{2} \) |
| 67 | \( 1 + 7.38T + 67T^{2} \) |
| 71 | \( 1 + 4.13T + 71T^{2} \) |
| 73 | \( 1 - 0.198T + 73T^{2} \) |
| 79 | \( 1 + 9.31T + 79T^{2} \) |
| 83 | \( 1 + 2.11T + 83T^{2} \) |
| 89 | \( 1 + 0.643T + 89T^{2} \) |
| 97 | \( 1 - 15.5T + 97T^{2} \) |
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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
−7.57578596414515520999365845632, −6.91659623692703277909706058044, −5.97462815987795671052236135432, −5.23452314926950735457747988910, −4.46213421745924742700851600003, −3.91153641726428256862010518725, −2.93764055795603929928375583585, −2.33014898706957958876203962485, −1.31565375221734365209350573474, 0,
1.31565375221734365209350573474, 2.33014898706957958876203962485, 2.93764055795603929928375583585, 3.91153641726428256862010518725, 4.46213421745924742700851600003, 5.23452314926950735457747988910, 5.97462815987795671052236135432, 6.91659623692703277909706058044, 7.57578596414515520999365845632