| L(s) = 1 | + 3-s + 5-s − 2·9-s − 5·11-s − 13-s + 15-s − 6·17-s + 4·19-s + 6·23-s − 4·25-s − 5·27-s + 29-s − 9·31-s − 5·33-s − 39-s − 8·41-s − 43-s − 2·45-s − 9·47-s − 7·49-s − 6·51-s + 9·53-s − 5·55-s + 4·57-s + 14·59-s − 10·61-s − 65-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s − 2/3·9-s − 1.50·11-s − 0.277·13-s + 0.258·15-s − 1.45·17-s + 0.917·19-s + 1.25·23-s − 4/5·25-s − 0.962·27-s + 0.185·29-s − 1.61·31-s − 0.870·33-s − 0.160·39-s − 1.24·41-s − 0.152·43-s − 0.298·45-s − 1.31·47-s − 49-s − 0.840·51-s + 1.23·53-s − 0.674·55-s + 0.529·57-s + 1.82·59-s − 1.28·61-s − 0.124·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 29 | \( 1 - T \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 9 T + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 - 14 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 17 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 4 T + p T^{2} \) |
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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
−8.881138385683193841802168431384, −8.113412320599282444813844622501, −7.40058900877586735707605147630, −6.51872387832910144837398766436, −5.39160479139122962668067738269, −5.01549072289869683857343789226, −3.58464413304689469477160283746, −2.72974150648470900016910402012, −1.96271232626592152421424128874, 0,
1.96271232626592152421424128874, 2.72974150648470900016910402012, 3.58464413304689469477160283746, 5.01549072289869683857343789226, 5.39160479139122962668067738269, 6.51872387832910144837398766436, 7.40058900877586735707605147630, 8.113412320599282444813844622501, 8.881138385683193841802168431384
