| L(s) = 1 | + 5-s − 2·11-s + 4·13-s + 2·17-s − 2·23-s + 25-s − 6·29-s − 4·31-s − 10·37-s − 2·41-s + 4·43-s − 8·47-s − 10·53-s − 2·55-s + 4·59-s − 8·61-s + 4·65-s + 8·67-s − 6·71-s − 4·73-s + 4·79-s + 4·83-s + 2·85-s − 10·89-s + 12·97-s − 14·101-s − 4·103-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.603·11-s + 1.10·13-s + 0.485·17-s − 0.417·23-s + 1/5·25-s − 1.11·29-s − 0.718·31-s − 1.64·37-s − 0.312·41-s + 0.609·43-s − 1.16·47-s − 1.37·53-s − 0.269·55-s + 0.520·59-s − 1.02·61-s + 0.496·65-s + 0.977·67-s − 0.712·71-s − 0.468·73-s + 0.450·79-s + 0.439·83-s + 0.216·85-s − 1.05·89-s + 1.21·97-s − 1.39·101-s − 0.394·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8820 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8820 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 12 T + p T^{2} \) |
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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
−7.42860709281337808833422377247, −6.70410746191297532773509276310, −5.92734363635461198172600157355, −5.47440495234875634265476665659, −4.70373701792905475760933628131, −3.69500645251956326263679346161, −3.19467507508242675366257447588, −2.06994310909687261486552154294, −1.38700491302511613161264275217, 0,
1.38700491302511613161264275217, 2.06994310909687261486552154294, 3.19467507508242675366257447588, 3.69500645251956326263679346161, 4.70373701792905475760933628131, 5.47440495234875634265476665659, 5.92734363635461198172600157355, 6.70410746191297532773509276310, 7.42860709281337808833422377247
