| L(s) = 1 | + 5-s − 3·9-s + 4·11-s − 6·13-s − 6·17-s + 19-s − 8·23-s + 25-s − 2·29-s + 2·37-s + 2·41-s − 4·43-s − 3·45-s + 8·47-s − 7·49-s − 6·53-s + 4·55-s + 4·59-s − 2·61-s − 6·65-s − 8·67-s − 8·71-s + 2·73-s + 8·79-s + 9·81-s − 4·83-s − 6·85-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 9-s + 1.20·11-s − 1.66·13-s − 1.45·17-s + 0.229·19-s − 1.66·23-s + 1/5·25-s − 0.371·29-s + 0.328·37-s + 0.312·41-s − 0.609·43-s − 0.447·45-s + 1.16·47-s − 49-s − 0.824·53-s + 0.539·55-s + 0.520·59-s − 0.256·61-s − 0.744·65-s − 0.977·67-s − 0.949·71-s + 0.234·73-s + 0.900·79-s + 81-s − 0.439·83-s − 0.650·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−9.205033424604682780071294890563, −8.367215276624493802870113616306, −7.42607731770411783498514204494, −6.53479363753959069873167494535, −5.88666763008921616620555990760, −4.87366194948920245730018828426, −4.01896931255446458307976642593, −2.73770469087314233186868460582, −1.89075429809912392568711002351, 0,
1.89075429809912392568711002351, 2.73770469087314233186868460582, 4.01896931255446458307976642593, 4.87366194948920245730018828426, 5.88666763008921616620555990760, 6.53479363753959069873167494535, 7.42607731770411783498514204494, 8.367215276624493802870113616306, 9.205033424604682780071294890563
