| L(s) = 1 | − 3-s + 0.144·5-s − 2.05·7-s + 9-s − 5.70·11-s − 4.46·13-s − 0.144·15-s + 4.26·17-s + 3.58·19-s + 2.05·21-s + 23-s − 4.97·25-s − 27-s + 29-s − 6.14·31-s + 5.70·33-s − 0.297·35-s + 9.53·37-s + 4.46·39-s − 7.46·41-s − 2.12·43-s + 0.144·45-s − 12.1·47-s − 2.77·49-s − 4.26·51-s + 6.19·53-s − 0.826·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.0647·5-s − 0.776·7-s + 0.333·9-s − 1.72·11-s − 1.23·13-s − 0.0374·15-s + 1.03·17-s + 0.821·19-s + 0.448·21-s + 0.208·23-s − 0.995·25-s − 0.192·27-s + 0.185·29-s − 1.10·31-s + 0.993·33-s − 0.0503·35-s + 1.56·37-s + 0.715·39-s − 1.16·41-s − 0.323·43-s + 0.0215·45-s − 1.76·47-s − 0.396·49-s − 0.596·51-s + 0.850·53-s − 0.111·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5913961051\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5913961051\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 - T \) |
| good | 5 | \( 1 - 0.144T + 5T^{2} \) |
| 7 | \( 1 + 2.05T + 7T^{2} \) |
| 11 | \( 1 + 5.70T + 11T^{2} \) |
| 13 | \( 1 + 4.46T + 13T^{2} \) |
| 17 | \( 1 - 4.26T + 17T^{2} \) |
| 19 | \( 1 - 3.58T + 19T^{2} \) |
| 31 | \( 1 + 6.14T + 31T^{2} \) |
| 37 | \( 1 - 9.53T + 37T^{2} \) |
| 41 | \( 1 + 7.46T + 41T^{2} \) |
| 43 | \( 1 + 2.12T + 43T^{2} \) |
| 47 | \( 1 + 12.1T + 47T^{2} \) |
| 53 | \( 1 - 6.19T + 53T^{2} \) |
| 59 | \( 1 + 13.2T + 59T^{2} \) |
| 61 | \( 1 - 5.14T + 61T^{2} \) |
| 67 | \( 1 + 0.811T + 67T^{2} \) |
| 71 | \( 1 - 11.4T + 71T^{2} \) |
| 73 | \( 1 + 13.7T + 73T^{2} \) |
| 79 | \( 1 - 1.51T + 79T^{2} \) |
| 83 | \( 1 + 13.5T + 83T^{2} \) |
| 89 | \( 1 - 4.10T + 89T^{2} \) |
| 97 | \( 1 + 4.29T + 97T^{2} \) |
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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
−7.67656921490044866822510316224, −7.26659229124331516510529499165, −6.41147799280412090841445421696, −5.53907754792289670439941034459, −5.27877071064723121845306008174, −4.46917435624086888915482172900, −3.31269419707289418913973949515, −2.81883826424261818423934646007, −1.77073283599473727817972811211, −0.37501078171997851013496673400,
0.37501078171997851013496673400, 1.77073283599473727817972811211, 2.81883826424261818423934646007, 3.31269419707289418913973949515, 4.46917435624086888915482172900, 5.27877071064723121845306008174, 5.53907754792289670439941034459, 6.41147799280412090841445421696, 7.26659229124331516510529499165, 7.67656921490044866822510316224
