| L(s) = 1 | + 2·7-s − 3·9-s + 6·11-s + 2·13-s + 6·17-s − 2·19-s + 6·23-s − 6·29-s − 4·31-s + 6·37-s − 2·41-s − 4·43-s − 10·47-s − 3·49-s + 2·53-s + 10·59-s + 10·61-s − 6·63-s + 4·67-s − 16·71-s + 6·73-s + 12·77-s + 9·81-s + 8·83-s + 6·89-s + 4·91-s − 2·97-s + ⋯ |
| L(s) = 1 | + 0.755·7-s − 9-s + 1.80·11-s + 0.554·13-s + 1.45·17-s − 0.458·19-s + 1.25·23-s − 1.11·29-s − 0.718·31-s + 0.986·37-s − 0.312·41-s − 0.609·43-s − 1.45·47-s − 3/7·49-s + 0.274·53-s + 1.30·59-s + 1.28·61-s − 0.755·63-s + 0.488·67-s − 1.89·71-s + 0.702·73-s + 1.36·77-s + 81-s + 0.878·83-s + 0.635·89-s + 0.419·91-s − 0.203·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.299070979\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.299070979\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 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
−8.664740514859947463487572146358, −8.059724653017145285250923659536, −7.15945501379979153094827252136, −6.36683816821148663491715471569, −5.64456854027065860896743359184, −4.90237225491692595850068450403, −3.81150237526906934474907121356, −3.25744467907735143549323839239, −1.88177651656706483466737260715, −0.984336091803417183888443825752,
0.984336091803417183888443825752, 1.88177651656706483466737260715, 3.25744467907735143549323839239, 3.81150237526906934474907121356, 4.90237225491692595850068450403, 5.64456854027065860896743359184, 6.36683816821148663491715471569, 7.15945501379979153094827252136, 8.059724653017145285250923659536, 8.664740514859947463487572146358
