| L(s) = 1 | + 2·3-s − 5-s − 2·7-s + 9-s + 2·13-s − 2·15-s − 6·17-s + 4·19-s − 4·21-s − 6·23-s + 25-s − 4·27-s + 6·29-s + 4·31-s + 2·35-s + 2·37-s + 4·39-s + 6·41-s + 10·43-s − 45-s + 6·47-s − 3·49-s − 12·51-s − 6·53-s + 8·57-s − 12·59-s + 2·61-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.447·5-s − 0.755·7-s + 1/3·9-s + 0.554·13-s − 0.516·15-s − 1.45·17-s + 0.917·19-s − 0.872·21-s − 1.25·23-s + 1/5·25-s − 0.769·27-s + 1.11·29-s + 0.718·31-s + 0.338·35-s + 0.328·37-s + 0.640·39-s + 0.937·41-s + 1.52·43-s − 0.149·45-s + 0.875·47-s − 3/7·49-s − 1.68·51-s − 0.824·53-s + 1.05·57-s − 1.56·59-s + 0.256·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.137082599\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.137082599\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 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
−14.20933451643817553759957219259, −13.57191933746624476808470409800, −12.41085901163162733003481941315, −11.10973086408237495845181170606, −9.683565298540966130897522415203, −8.740293324354028093320034492248, −7.71273110823499542846308181544, −6.27087624192875571051851265258, −4.12250433368686324236236368171, −2.76929890617261215013507568311,
2.76929890617261215013507568311, 4.12250433368686324236236368171, 6.27087624192875571051851265258, 7.71273110823499542846308181544, 8.740293324354028093320034492248, 9.683565298540966130897522415203, 11.10973086408237495845181170606, 12.41085901163162733003481941315, 13.57191933746624476808470409800, 14.20933451643817553759957219259
