| L(s) = 1 | + 2-s + 3-s + 4-s + 6-s + 7-s + 8-s + 9-s − 11-s + 12-s + 2·13-s + 14-s + 16-s + 18-s + 2·19-s + 21-s − 22-s + 24-s − 5·25-s + 2·26-s + 27-s + 28-s − 6·29-s + 2·31-s + 32-s − 33-s + 36-s + 2·37-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s − 0.301·11-s + 0.288·12-s + 0.554·13-s + 0.267·14-s + 1/4·16-s + 0.235·18-s + 0.458·19-s + 0.218·21-s − 0.213·22-s + 0.204·24-s − 25-s + 0.392·26-s + 0.192·27-s + 0.188·28-s − 1.11·29-s + 0.359·31-s + 0.176·32-s − 0.174·33-s + 1/6·36-s + 0.328·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.627206274\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.627206274\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−11.21492657602357137923632233433, −10.22218241237583663002107615274, −9.260014518115933312672207760363, −8.158830011188341869727925681893, −7.46173924472506058951788867760, −6.27935085429088375420913933534, −5.25821839364078131696035606890, −4.14440200645550446493587121840, −3.12443402203060754923261311981, −1.77631258837245971519374380176,
1.77631258837245971519374380176, 3.12443402203060754923261311981, 4.14440200645550446493587121840, 5.25821839364078131696035606890, 6.27935085429088375420913933534, 7.46173924472506058951788867760, 8.158830011188341869727925681893, 9.260014518115933312672207760363, 10.22218241237583663002107615274, 11.21492657602357137923632233433
