| L(s) = 1 | − 5-s − 2·11-s − 2·13-s + 6·17-s + 2·19-s − 8·23-s + 25-s − 4·29-s + 2·31-s + 2·37-s − 6·41-s − 4·43-s − 4·47-s − 2·53-s + 2·55-s + 4·59-s + 12·61-s + 2·65-s + 16·67-s − 6·71-s + 14·73-s − 8·79-s − 4·83-s − 6·85-s − 10·89-s − 2·95-s + 10·97-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.603·11-s − 0.554·13-s + 1.45·17-s + 0.458·19-s − 1.66·23-s + 1/5·25-s − 0.742·29-s + 0.359·31-s + 0.328·37-s − 0.937·41-s − 0.609·43-s − 0.583·47-s − 0.274·53-s + 0.269·55-s + 0.520·59-s + 1.53·61-s + 0.248·65-s + 1.95·67-s − 0.712·71-s + 1.63·73-s − 0.900·79-s − 0.439·83-s − 0.650·85-s − 1.05·89-s − 0.205·95-s + 1.01·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 141120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 141120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 12 T + p T^{2} \) |
| 67 | \( 1 - 16 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−13.73062445860843, −12.94930839306257, −12.80310159818335, −12.09518694148652, −11.76646975651319, −11.42037739337797, −10.73906965524869, −10.11161635664441, −9.841320283118704, −9.555578233672815, −8.590616832005509, −8.196519415228351, −7.854387532706753, −7.359278826399214, −6.830837379906763, −6.202968038753857, −5.540791444702465, −5.237543307335419, −4.667048246162107, −3.823187136787466, −3.600394693012078, −2.864711090194002, −2.247436373999763, −1.586860679421500, −0.7420644429603934, 0,
0.7420644429603934, 1.586860679421500, 2.247436373999763, 2.864711090194002, 3.600394693012078, 3.823187136787466, 4.667048246162107, 5.237543307335419, 5.540791444702465, 6.202968038753857, 6.830837379906763, 7.359278826399214, 7.854387532706753, 8.196519415228351, 8.590616832005509, 9.555578233672815, 9.841320283118704, 10.11161635664441, 10.73906965524869, 11.42037739337797, 11.76646975651319, 12.09518694148652, 12.80310159818335, 12.94930839306257, 13.73062445860843
