| L(s) = 1 | + 2·2-s + 3·3-s + 4·4-s − 5·5-s + 6·6-s + 8·8-s + 9·9-s − 10·10-s + 16·11-s + 12·12-s − 58·13-s − 15·15-s + 16·16-s − 34·17-s + 18·18-s − 64·19-s − 20·20-s + 32·22-s − 16·23-s + 24·24-s + 25·25-s − 116·26-s + 27·27-s + 62·29-s − 30·30-s − 60·31-s + 32·32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s + 0.353·8-s + 1/3·9-s − 0.316·10-s + 0.438·11-s + 0.288·12-s − 1.23·13-s − 0.258·15-s + 1/4·16-s − 0.485·17-s + 0.235·18-s − 0.772·19-s − 0.223·20-s + 0.310·22-s − 0.145·23-s + 0.204·24-s + 1/5·25-s − 0.874·26-s + 0.192·27-s + 0.397·29-s − 0.182·30-s − 0.347·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p T \) |
| 3 | \( 1 - p T \) |
| 5 | \( 1 + p T \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 - 16 T + p^{3} T^{2} \) |
| 13 | \( 1 + 58 T + p^{3} T^{2} \) |
| 17 | \( 1 + 2 p T + p^{3} T^{2} \) |
| 19 | \( 1 + 64 T + p^{3} T^{2} \) |
| 23 | \( 1 + 16 T + p^{3} T^{2} \) |
| 29 | \( 1 - 62 T + p^{3} T^{2} \) |
| 31 | \( 1 + 60 T + p^{3} T^{2} \) |
| 37 | \( 1 - 150 T + p^{3} T^{2} \) |
| 41 | \( 1 + 474 T + p^{3} T^{2} \) |
| 43 | \( 1 + 292 T + p^{3} T^{2} \) |
| 47 | \( 1 + 240 T + p^{3} T^{2} \) |
| 53 | \( 1 + 662 T + p^{3} T^{2} \) |
| 59 | \( 1 - 324 T + p^{3} T^{2} \) |
| 61 | \( 1 - 514 T + p^{3} T^{2} \) |
| 67 | \( 1 + 372 T + p^{3} T^{2} \) |
| 71 | \( 1 + 412 T + p^{3} T^{2} \) |
| 73 | \( 1 - 770 T + p^{3} T^{2} \) |
| 79 | \( 1 + 560 T + p^{3} T^{2} \) |
| 83 | \( 1 - 852 T + p^{3} T^{2} \) |
| 89 | \( 1 + 1466 T + p^{3} T^{2} \) |
| 97 | \( 1 - 178 T + p^{3} T^{2} \) |
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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
−8.589730389785564862434571112994, −7.926730431841472521606258291997, −6.98976406653486151230713505786, −6.45200886319131444785285829263, −5.13966299632155930883509330600, −4.46813671607493311692785042886, −3.59471501376483369452689650355, −2.66012898788104673699929867276, −1.68998485983365503426229209025, 0,
1.68998485983365503426229209025, 2.66012898788104673699929867276, 3.59471501376483369452689650355, 4.46813671607493311692785042886, 5.13966299632155930883509330600, 6.45200886319131444785285829263, 6.98976406653486151230713505786, 7.926730431841472521606258291997, 8.589730389785564862434571112994
