| L(s) = 1 | − 2·3-s + 9-s − 2·13-s + 6·17-s − 2·23-s + 4·27-s + 8·29-s − 4·31-s + 2·37-s + 4·39-s + 4·41-s + 4·43-s + 2·47-s − 7·49-s − 12·51-s + 10·53-s − 8·61-s − 2·67-s + 4·69-s − 8·71-s − 10·73-s − 4·79-s − 11·81-s − 12·83-s − 16·87-s + 6·89-s + 8·93-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 1/3·9-s − 0.554·13-s + 1.45·17-s − 0.417·23-s + 0.769·27-s + 1.48·29-s − 0.718·31-s + 0.328·37-s + 0.640·39-s + 0.624·41-s + 0.609·43-s + 0.291·47-s − 49-s − 1.68·51-s + 1.37·53-s − 1.02·61-s − 0.244·67-s + 0.481·69-s − 0.949·71-s − 1.17·73-s − 0.450·79-s − 1.22·81-s − 1.31·83-s − 1.71·87-s + 0.635·89-s + 0.829·93-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 48400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−14.71550360546996, −14.29952570290851, −13.94328352471249, −13.10404832887814, −12.58428753027272, −12.17807066476576, −11.77628812635731, −11.31338646985470, −10.66900890834437, −10.12889367157685, −9.939349210307111, −9.037857887086184, −8.588278945631763, −7.739237641901566, −7.462867070531414, −6.721760348148433, −6.107384232364592, −5.708584646413711, −5.187916210125570, −4.580252821566093, −4.001749597584414, −3.077940055128761, −2.606975394423538, −1.515947356373888, −0.8621739977240289, 0,
0.8621739977240289, 1.515947356373888, 2.606975394423538, 3.077940055128761, 4.001749597584414, 4.580252821566093, 5.187916210125570, 5.708584646413711, 6.107384232364592, 6.721760348148433, 7.462867070531414, 7.739237641901566, 8.588278945631763, 9.037857887086184, 9.939349210307111, 10.12889367157685, 10.66900890834437, 11.31338646985470, 11.77628812635731, 12.17807066476576, 12.58428753027272, 13.10404832887814, 13.94328352471249, 14.29952570290851, 14.71550360546996
