| L(s) = 1 | + 3-s − 5-s + 2·7-s + 9-s − 2·11-s + 6·13-s − 15-s − 6·17-s + 6·19-s + 2·21-s − 2·23-s + 25-s + 27-s + 2·29-s + 4·31-s − 2·33-s − 2·35-s + 10·37-s + 6·39-s − 2·41-s − 8·43-s − 45-s + 6·47-s − 3·49-s − 6·51-s + 6·53-s + 2·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s + 0.755·7-s + 1/3·9-s − 0.603·11-s + 1.66·13-s − 0.258·15-s − 1.45·17-s + 1.37·19-s + 0.436·21-s − 0.417·23-s + 1/5·25-s + 0.192·27-s + 0.371·29-s + 0.718·31-s − 0.348·33-s − 0.338·35-s + 1.64·37-s + 0.960·39-s − 0.312·41-s − 1.21·43-s − 0.149·45-s + 0.875·47-s − 3/7·49-s − 0.840·51-s + 0.824·53-s + 0.269·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.286967511\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.286967511\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 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
−9.009782191031173971561105640243, −8.300454644250546298995728078650, −7.921959866288433993862715383143, −6.94048247529211023854157343618, −6.07380284801590675671363852873, −5.00646292439135152348096305314, −4.21182094038261900638983310748, −3.34183055256449834784989322034, −2.28231008886779719005891093166, −1.05573399756661793291429489696,
1.05573399756661793291429489696, 2.28231008886779719005891093166, 3.34183055256449834784989322034, 4.21182094038261900638983310748, 5.00646292439135152348096305314, 6.07380284801590675671363852873, 6.94048247529211023854157343618, 7.921959866288433993862715383143, 8.300454644250546298995728078650, 9.009782191031173971561105640243
