| L(s) = 1 | + 0.678·3-s − 2.77·5-s + 1.71·7-s − 2.53·9-s + 0.585·11-s − 6.56·13-s − 1.88·15-s + 0.228·17-s − 5.57·19-s + 1.16·21-s + 4.63·23-s + 2.67·25-s − 3.76·27-s − 6.25·29-s + 7.04·31-s + 0.397·33-s − 4.74·35-s − 11.6·37-s − 4.45·39-s + 2.26·41-s − 3.41·43-s + 7.03·45-s − 6.83·47-s − 4.06·49-s + 0.154·51-s + 9.69·53-s − 1.62·55-s + ⋯ |
| L(s) = 1 | + 0.391·3-s − 1.23·5-s + 0.647·7-s − 0.846·9-s + 0.176·11-s − 1.82·13-s − 0.485·15-s + 0.0553·17-s − 1.27·19-s + 0.253·21-s + 0.966·23-s + 0.535·25-s − 0.723·27-s − 1.16·29-s + 1.26·31-s + 0.0692·33-s − 0.802·35-s − 1.92·37-s − 0.713·39-s + 0.354·41-s − 0.520·43-s + 1.04·45-s − 0.997·47-s − 0.580·49-s + 0.0216·51-s + 1.33·53-s − 0.218·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 668 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 668 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 167 | \( 1 - T \) |
| good | 3 | \( 1 - 0.678T + 3T^{2} \) |
| 5 | \( 1 + 2.77T + 5T^{2} \) |
| 7 | \( 1 - 1.71T + 7T^{2} \) |
| 11 | \( 1 - 0.585T + 11T^{2} \) |
| 13 | \( 1 + 6.56T + 13T^{2} \) |
| 17 | \( 1 - 0.228T + 17T^{2} \) |
| 19 | \( 1 + 5.57T + 19T^{2} \) |
| 23 | \( 1 - 4.63T + 23T^{2} \) |
| 29 | \( 1 + 6.25T + 29T^{2} \) |
| 31 | \( 1 - 7.04T + 31T^{2} \) |
| 37 | \( 1 + 11.6T + 37T^{2} \) |
| 41 | \( 1 - 2.26T + 41T^{2} \) |
| 43 | \( 1 + 3.41T + 43T^{2} \) |
| 47 | \( 1 + 6.83T + 47T^{2} \) |
| 53 | \( 1 - 9.69T + 53T^{2} \) |
| 59 | \( 1 - 9.29T + 59T^{2} \) |
| 61 | \( 1 + 2.23T + 61T^{2} \) |
| 67 | \( 1 + 5.40T + 67T^{2} \) |
| 71 | \( 1 + 10.7T + 71T^{2} \) |
| 73 | \( 1 - 14.0T + 73T^{2} \) |
| 79 | \( 1 + 9.29T + 79T^{2} \) |
| 83 | \( 1 - 14.0T + 83T^{2} \) |
| 89 | \( 1 - 2.45T + 89T^{2} \) |
| 97 | \( 1 - 4.76T + 97T^{2} \) |
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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
−10.11951500641762171816957416434, −8.982406274443790524192168586464, −8.301608297766906297813323295662, −7.60014951678672171676540852358, −6.78174446001659685766325114576, −5.28641371666727438391264613380, −4.48680623833834489173220618437, −3.37879658772284456158949981901, −2.22368035850675107672141803074, 0,
2.22368035850675107672141803074, 3.37879658772284456158949981901, 4.48680623833834489173220618437, 5.28641371666727438391264613380, 6.78174446001659685766325114576, 7.60014951678672171676540852358, 8.301608297766906297813323295662, 8.982406274443790524192168586464, 10.11951500641762171816957416434
