| L(s) = 1 | − 0.307·2-s − 1.90·4-s + 0.988·5-s + 7-s + 1.20·8-s − 0.303·10-s − 2.00·11-s − 0.289·13-s − 0.307·14-s + 3.44·16-s − 0.186·17-s + 0.852·19-s − 1.88·20-s + 0.616·22-s + 3.33·23-s − 4.02·25-s + 0.0890·26-s − 1.90·28-s + 4.37·29-s − 5.73·31-s − 3.45·32-s + 0.0572·34-s + 0.988·35-s − 7.05·37-s − 0.261·38-s + 1.18·40-s − 3.81·41-s + ⋯ |
| L(s) = 1 | − 0.217·2-s − 0.952·4-s + 0.442·5-s + 0.377·7-s + 0.424·8-s − 0.0960·10-s − 0.605·11-s − 0.0803·13-s − 0.0821·14-s + 0.860·16-s − 0.0451·17-s + 0.195·19-s − 0.421·20-s + 0.131·22-s + 0.695·23-s − 0.804·25-s + 0.0174·26-s − 0.360·28-s + 0.812·29-s − 1.03·31-s − 0.611·32-s + 0.00981·34-s + 0.167·35-s − 1.15·37-s − 0.0424·38-s + 0.187·40-s − 0.596·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 127 | \( 1 + T \) |
| good | 2 | \( 1 + 0.307T + 2T^{2} \) |
| 5 | \( 1 - 0.988T + 5T^{2} \) |
| 11 | \( 1 + 2.00T + 11T^{2} \) |
| 13 | \( 1 + 0.289T + 13T^{2} \) |
| 17 | \( 1 + 0.186T + 17T^{2} \) |
| 19 | \( 1 - 0.852T + 19T^{2} \) |
| 23 | \( 1 - 3.33T + 23T^{2} \) |
| 29 | \( 1 - 4.37T + 29T^{2} \) |
| 31 | \( 1 + 5.73T + 31T^{2} \) |
| 37 | \( 1 + 7.05T + 37T^{2} \) |
| 41 | \( 1 + 3.81T + 41T^{2} \) |
| 43 | \( 1 - 9.06T + 43T^{2} \) |
| 47 | \( 1 + 1.34T + 47T^{2} \) |
| 53 | \( 1 + 3.75T + 53T^{2} \) |
| 59 | \( 1 - 4.26T + 59T^{2} \) |
| 61 | \( 1 - 3.08T + 61T^{2} \) |
| 67 | \( 1 + 10.8T + 67T^{2} \) |
| 71 | \( 1 + 13.0T + 71T^{2} \) |
| 73 | \( 1 - 14.9T + 73T^{2} \) |
| 79 | \( 1 - 16.7T + 79T^{2} \) |
| 83 | \( 1 + 15.1T + 83T^{2} \) |
| 89 | \( 1 - 0.526T + 89T^{2} \) |
| 97 | \( 1 - 6.83T + 97T^{2} \) |
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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
−7.61101355213255224160276775080, −6.93809730268895159346427513449, −5.91439686074406098642649852177, −5.32392641924756193435717652293, −4.79027192212279781619500303321, −3.97196051223100766481451386034, −3.14473485645889243656324651941, −2.11916747369891743343312801012, −1.18339700259551294861121555385, 0,
1.18339700259551294861121555385, 2.11916747369891743343312801012, 3.14473485645889243656324651941, 3.97196051223100766481451386034, 4.79027192212279781619500303321, 5.32392641924756193435717652293, 5.91439686074406098642649852177, 6.93809730268895159346427513449, 7.61101355213255224160276775080
