| L(s) = 1 | + 5-s − 2·7-s − 4·13-s + 2·17-s − 19-s + 6·23-s + 25-s + 2·29-s − 2·35-s − 4·37-s − 2·41-s + 2·43-s + 6·47-s − 3·49-s + 12·53-s + 4·59-s + 6·61-s − 4·65-s + 12·67-s − 2·73-s + 6·83-s + 2·85-s + 14·89-s + 8·91-s − 95-s + 8·97-s − 10·101-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.755·7-s − 1.10·13-s + 0.485·17-s − 0.229·19-s + 1.25·23-s + 1/5·25-s + 0.371·29-s − 0.338·35-s − 0.657·37-s − 0.312·41-s + 0.304·43-s + 0.875·47-s − 3/7·49-s + 1.64·53-s + 0.520·59-s + 0.768·61-s − 0.496·65-s + 1.46·67-s − 0.234·73-s + 0.658·83-s + 0.216·85-s + 1.48·89-s + 0.838·91-s − 0.102·95-s + 0.812·97-s − 0.995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3420 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.684506805\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.684506805\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−8.781070552959257577208804589983, −7.77202455346981741431783987789, −7.01222136762435827798054936680, −6.49124677439142411120091764623, −5.48601239916056582187808892036, −4.95569243410605872203572415004, −3.84968448773156771376028721915, −2.94968445495291061004688085463, −2.16615349123070617018883400977, −0.75756018194514963077733170638,
0.75756018194514963077733170638, 2.16615349123070617018883400977, 2.94968445495291061004688085463, 3.84968448773156771376028721915, 4.95569243410605872203572415004, 5.48601239916056582187808892036, 6.49124677439142411120091764623, 7.01222136762435827798054936680, 7.77202455346981741431783987789, 8.781070552959257577208804589983
