| L(s) = 1 | − 2-s + 4-s + 5-s + 2·7-s − 8-s − 10-s − 3·11-s + 13-s − 2·14-s + 16-s − 7·19-s + 20-s + 3·22-s + 4·23-s + 25-s − 26-s + 2·28-s − 4·29-s + 11·31-s − 32-s + 2·35-s + 9·37-s + 7·38-s − 40-s + 4·41-s − 11·43-s − 3·44-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.447·5-s + 0.755·7-s − 0.353·8-s − 0.316·10-s − 0.904·11-s + 0.277·13-s − 0.534·14-s + 1/4·16-s − 1.60·19-s + 0.223·20-s + 0.639·22-s + 0.834·23-s + 1/5·25-s − 0.196·26-s + 0.377·28-s − 0.742·29-s + 1.97·31-s − 0.176·32-s + 0.338·35-s + 1.47·37-s + 1.13·38-s − 0.158·40-s + 0.624·41-s − 1.67·43-s − 0.452·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 338130 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 338130 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.797213036\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.797213036\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 - 11 T + p T^{2} \) |
| 37 | \( 1 - 9 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 + 11 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 5 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 11 T + p T^{2} \) |
| 89 | \( 1 - 9 T + p T^{2} \) |
| 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
−12.46930233360102, −12.19016830537145, −11.40326725453646, −11.12611638300213, −10.80414293366280, −10.24881642739258, −9.966295259735348, −9.312074607694991, −8.948070246186308, −8.345061427057177, −8.093076529250086, −7.690899983220051, −7.089048273684099, −6.477818832594608, −6.153587543185266, −5.687805593141279, −4.909885505960629, −4.665115595743666, −4.118780244940204, −3.199131096944761, −2.800221117204321, −2.195682230902777, −1.750243653076008, −1.086707605780606, −0.4169759059033768,
0.4169759059033768, 1.086707605780606, 1.750243653076008, 2.195682230902777, 2.800221117204321, 3.199131096944761, 4.118780244940204, 4.665115595743666, 4.909885505960629, 5.687805593141279, 6.153587543185266, 6.477818832594608, 7.089048273684099, 7.690899983220051, 8.093076529250086, 8.345061427057177, 8.948070246186308, 9.312074607694991, 9.966295259735348, 10.24881642739258, 10.80414293366280, 11.12611638300213, 11.40326725453646, 12.19016830537145, 12.46930233360102
