| L(s) = 1 | − 3-s + 9-s + 2·11-s + 3·13-s − 6·17-s + 7·19-s + 6·23-s − 27-s − 2·29-s + 5·31-s − 2·33-s + 10·37-s − 3·39-s − 12·41-s + 3·43-s + 10·47-s + 6·51-s − 7·57-s + 6·59-s + 13·61-s + 7·67-s − 6·69-s − 4·71-s + 6·73-s − 8·79-s + 81-s + 6·83-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/3·9-s + 0.603·11-s + 0.832·13-s − 1.45·17-s + 1.60·19-s + 1.25·23-s − 0.192·27-s − 0.371·29-s + 0.898·31-s − 0.348·33-s + 1.64·37-s − 0.480·39-s − 1.87·41-s + 0.457·43-s + 1.45·47-s + 0.840·51-s − 0.927·57-s + 0.781·59-s + 1.66·61-s + 0.855·67-s − 0.722·69-s − 0.474·71-s + 0.702·73-s − 0.900·79-s + 1/9·81-s + 0.658·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.372783776\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.372783776\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 7 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 - 3 T + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 13 T + p T^{2} \) |
| 67 | \( 1 - 7 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 16 T + p T^{2} \) |
| 97 | \( 1 - 7 T + p T^{2} \) |
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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
−15.34058933771130, −14.70241688410515, −13.94174637621418, −13.57703251506188, −13.05038446864682, −12.56810987864251, −11.69976613893606, −11.41709893713065, −11.13727659204054, −10.30815369784832, −9.818068426784146, −9.120552688877005, −8.768474584914943, −8.056433055756345, −7.276678879425808, −6.799064784322507, −6.341680814385732, −5.592743154128062, −5.097711874021555, −4.356403118857061, −3.811272177272778, −3.030159336031480, −2.241292497737979, −1.228756340865216, −0.7109614967187371,
0.7109614967187371, 1.228756340865216, 2.241292497737979, 3.030159336031480, 3.811272177272778, 4.356403118857061, 5.097711874021555, 5.592743154128062, 6.341680814385732, 6.799064784322507, 7.276678879425808, 8.056433055756345, 8.768474584914943, 9.120552688877005, 9.818068426784146, 10.30815369784832, 11.13727659204054, 11.41709893713065, 11.69976613893606, 12.56810987864251, 13.05038446864682, 13.57703251506188, 13.94174637621418, 14.70241688410515, 15.34058933771130
