| L(s) = 1 | + 2.73·3-s − 2.73·5-s + 7-s + 4.46·9-s − 5.46·11-s − 6.73·13-s − 7.46·15-s + 2·17-s − 1.26·19-s + 2.73·21-s − 3.46·23-s + 2.46·25-s + 3.99·27-s + 1.46·29-s − 4·31-s − 14.9·33-s − 2.73·35-s + 1.46·37-s − 18.3·39-s + 2·41-s − 5.46·43-s − 12.1·45-s − 2.92·47-s + 49-s + 5.46·51-s − 12·53-s + 14.9·55-s + ⋯ |
| L(s) = 1 | + 1.57·3-s − 1.22·5-s + 0.377·7-s + 1.48·9-s − 1.64·11-s − 1.86·13-s − 1.92·15-s + 0.485·17-s − 0.290·19-s + 0.596·21-s − 0.722·23-s + 0.492·25-s + 0.769·27-s + 0.271·29-s − 0.718·31-s − 2.59·33-s − 0.461·35-s + 0.240·37-s − 2.94·39-s + 0.312·41-s − 0.833·43-s − 1.81·45-s − 0.427·47-s + 0.142·49-s + 0.765·51-s − 1.64·53-s + 2.01·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 3 | \( 1 - 2.73T + 3T^{2} \) |
| 5 | \( 1 + 2.73T + 5T^{2} \) |
| 11 | \( 1 + 5.46T + 11T^{2} \) |
| 13 | \( 1 + 6.73T + 13T^{2} \) |
| 17 | \( 1 - 2T + 17T^{2} \) |
| 19 | \( 1 + 1.26T + 19T^{2} \) |
| 23 | \( 1 + 3.46T + 23T^{2} \) |
| 29 | \( 1 - 1.46T + 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 - 1.46T + 37T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 + 5.46T + 43T^{2} \) |
| 47 | \( 1 + 2.92T + 47T^{2} \) |
| 53 | \( 1 + 12T + 53T^{2} \) |
| 59 | \( 1 - 9.66T + 59T^{2} \) |
| 61 | \( 1 - 11.1T + 61T^{2} \) |
| 67 | \( 1 - 8T + 67T^{2} \) |
| 71 | \( 1 - 2.92T + 71T^{2} \) |
| 73 | \( 1 + 12.9T + 73T^{2} \) |
| 79 | \( 1 - 10.9T + 79T^{2} \) |
| 83 | \( 1 + 5.66T + 83T^{2} \) |
| 89 | \( 1 + 11.8T + 89T^{2} \) |
| 97 | \( 1 - 8.92T + 97T^{2} \) |
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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
−8.592667491135358295514160456258, −7.925891354749374689580505613546, −7.74227812686545930648551882902, −7.00959202411629396276321208044, −5.34000762845473403855446184716, −4.59223074162802142399922523261, −3.68145721929282515553816943297, −2.79696015752342200606741578769, −2.11646421928625919274913670418, 0,
2.11646421928625919274913670418, 2.79696015752342200606741578769, 3.68145721929282515553816943297, 4.59223074162802142399922523261, 5.34000762845473403855446184716, 7.00959202411629396276321208044, 7.74227812686545930648551882902, 7.925891354749374689580505613546, 8.592667491135358295514160456258
