| L(s) = 1 | + 2-s + 3-s + 4-s + 6-s − 7-s + 8-s + 9-s + 2·11-s + 12-s + 13-s − 14-s + 16-s − 17-s + 18-s + 4·19-s − 21-s + 2·22-s + 7·23-s + 24-s + 26-s + 27-s − 28-s + 29-s + 3·31-s + 32-s + 2·33-s − 34-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 + 0.603·11-s + 0.288·12-s + 0.277·13-s − 0.267·14-s + 1/4·16-s − 0.242·17-s + 0.235·18-s + 0.917·19-s − 0.218·21-s + 0.426·22-s + 1.45·23-s + 0.204·24-s + 0.196·26-s + 0.192·27-s − 0.188·28-s + 0.185·29-s + 0.538·31-s + 0.176·32-s + 0.348·33-s − 0.171·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.138982702\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.138982702\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 7 T + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 11 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 + 3 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−9.883845251580328817276617277998, −9.074194897731522107608923402365, −8.297458249941517050722181057396, −7.16320504908367225259765767837, −6.65245898569751299827159930598, −5.53310237949696162438796572315, −4.59186140813815219977393447102, −3.54905228293585120115308652444, −2.83900221572586575473984990135, −1.40149230437420693326704510543,
1.40149230437420693326704510543, 2.83900221572586575473984990135, 3.54905228293585120115308652444, 4.59186140813815219977393447102, 5.53310237949696162438796572315, 6.65245898569751299827159930598, 7.16320504908367225259765767837, 8.297458249941517050722181057396, 9.074194897731522107608923402365, 9.883845251580328817276617277998
