| L(s) = 1 | + 3-s − 2·7-s − 2·9-s + 3·11-s + 6·13-s + 3·17-s + 5·19-s − 2·21-s + 4·23-s − 5·27-s + 4·29-s + 2·31-s + 3·33-s + 4·37-s + 6·39-s + 9·41-s + 4·43-s − 3·49-s + 3·51-s + 2·53-s + 5·57-s + 4·59-s − 61-s + 4·63-s + 5·67-s + 4·69-s + 14·71-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.755·7-s − 2/3·9-s + 0.904·11-s + 1.66·13-s + 0.727·17-s + 1.14·19-s − 0.436·21-s + 0.834·23-s − 0.962·27-s + 0.742·29-s + 0.359·31-s + 0.522·33-s + 0.657·37-s + 0.960·39-s + 1.40·41-s + 0.609·43-s − 3/7·49-s + 0.420·51-s + 0.274·53-s + 0.662·57-s + 0.520·59-s − 0.128·61-s + 0.503·63-s + 0.610·67-s + 0.481·69-s + 1.66·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 24400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 24400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.483580001\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.483580001\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 61 | \( 1 + T \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 - 14 T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 + T + p T^{2} \) |
| 89 | \( 1 - T + p T^{2} \) |
| 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
−15.54492850893401, −14.71427733110048, −14.18906873371606, −13.98075622498237, −13.26244519318844, −12.87559976801598, −12.12901029561195, −11.56929510033443, −11.13490028393596, −10.52014862831611, −9.691904634286691, −9.260544809212469, −8.949892030394151, −8.066326605444281, −7.887266216361327, −6.773769807956859, −6.505270979010594, −5.729462471472845, −5.315123091327256, −4.136170096883348, −3.709071224821560, −3.063727460095448, −2.590410133885088, −1.315441004501026, −0.8229598394164941,
0.8229598394164941, 1.315441004501026, 2.590410133885088, 3.063727460095448, 3.709071224821560, 4.136170096883348, 5.315123091327256, 5.729462471472845, 6.505270979010594, 6.773769807956859, 7.887266216361327, 8.066326605444281, 8.949892030394151, 9.260544809212469, 9.691904634286691, 10.52014862831611, 11.13490028393596, 11.56929510033443, 12.12901029561195, 12.87559976801598, 13.26244519318844, 13.98075622498237, 14.18906873371606, 14.71427733110048, 15.54492850893401
