| L(s) = 1 | + 2-s − 3-s + 4-s + 0.199·5-s − 6-s − 0.938·7-s + 8-s + 9-s + 0.199·10-s − 11-s − 12-s − 0.930·13-s − 0.938·14-s − 0.199·15-s + 16-s − 2.81·17-s + 18-s − 2.14·19-s + 0.199·20-s + 0.938·21-s − 22-s + 7.96·23-s − 24-s − 4.96·25-s − 0.930·26-s − 27-s − 0.938·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.0890·5-s − 0.408·6-s − 0.354·7-s + 0.353·8-s + 0.333·9-s + 0.0629·10-s − 0.301·11-s − 0.288·12-s − 0.258·13-s − 0.250·14-s − 0.0514·15-s + 0.250·16-s − 0.683·17-s + 0.235·18-s − 0.491·19-s + 0.0445·20-s + 0.204·21-s − 0.213·22-s + 1.66·23-s − 0.204·24-s − 0.992·25-s − 0.182·26-s − 0.192·27-s − 0.177·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 61 | \( 1 - T \) |
| good | 5 | \( 1 - 0.199T + 5T^{2} \) |
| 7 | \( 1 + 0.938T + 7T^{2} \) |
| 13 | \( 1 + 0.930T + 13T^{2} \) |
| 17 | \( 1 + 2.81T + 17T^{2} \) |
| 19 | \( 1 + 2.14T + 19T^{2} \) |
| 23 | \( 1 - 7.96T + 23T^{2} \) |
| 29 | \( 1 - 2.70T + 29T^{2} \) |
| 31 | \( 1 + 1.54T + 31T^{2} \) |
| 37 | \( 1 + 3.07T + 37T^{2} \) |
| 41 | \( 1 - 0.0554T + 41T^{2} \) |
| 43 | \( 1 + 9.33T + 43T^{2} \) |
| 47 | \( 1 - 1.16T + 47T^{2} \) |
| 53 | \( 1 + 2.09T + 53T^{2} \) |
| 59 | \( 1 + 12.0T + 59T^{2} \) |
| 67 | \( 1 - 13.9T + 67T^{2} \) |
| 71 | \( 1 - 5.25T + 71T^{2} \) |
| 73 | \( 1 - 1.42T + 73T^{2} \) |
| 79 | \( 1 + 13.6T + 79T^{2} \) |
| 83 | \( 1 + 13.4T + 83T^{2} \) |
| 89 | \( 1 + 12.9T + 89T^{2} \) |
| 97 | \( 1 - 3.31T + 97T^{2} \) |
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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
−7.958682149369735116414967352797, −6.98050598111704135855309060784, −6.63514208256575157255538292668, −5.76742349390376578210200002482, −5.05932316351530769167724171011, −4.44071384247411066180378883859, −3.46989814236019027658431186077, −2.60813869936107821381264463025, −1.53643018894913713552099416987, 0,
1.53643018894913713552099416987, 2.60813869936107821381264463025, 3.46989814236019027658431186077, 4.44071384247411066180378883859, 5.05932316351530769167724171011, 5.76742349390376578210200002482, 6.63514208256575157255538292668, 6.98050598111704135855309060784, 7.958682149369735116414967352797
