| L(s) = 1 | − 2-s + 3-s + 4-s − 6-s + 7-s − 8-s + 9-s + 11-s + 12-s + 6·13-s − 14-s + 16-s + 17-s − 18-s + 6·19-s + 21-s − 22-s + 3·23-s − 24-s − 5·25-s − 6·26-s + 27-s + 28-s + 6·29-s − 3·31-s − 32-s + 33-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 + 1.66·13-s − 0.267·14-s + 1/4·16-s + 0.242·17-s − 0.235·18-s + 1.37·19-s + 0.218·21-s − 0.213·22-s + 0.625·23-s − 0.204·24-s − 25-s − 1.17·26-s + 0.192·27-s + 0.188·28-s + 1.11·29-s − 0.538·31-s − 0.176·32-s + 0.174·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225318 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225318 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.651827181\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.651827181\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| 47 | \( 1 \) |
| good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 9 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 7 T + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
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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
−13.07748616260286, −12.34996788451842, −11.93540816417196, −11.46630369166040, −11.07328228997050, −10.60131995978503, −10.04967436473114, −9.645476096647777, −9.047549079045157, −8.817740591873026, −8.238719773726350, −7.873720832684077, −7.398222160887466, −6.844617717107724, −6.382844551665795, −5.688988167853393, −5.434635826643981, −4.548235600275765, −4.042826840339189, −3.392996477205881, −3.092953795270685, −2.356200819635243, −1.561519199679935, −1.264697517698102, −0.6171433564841832,
0.6171433564841832, 1.264697517698102, 1.561519199679935, 2.356200819635243, 3.092953795270685, 3.392996477205881, 4.042826840339189, 4.548235600275765, 5.434635826643981, 5.688988167853393, 6.382844551665795, 6.844617717107724, 7.398222160887466, 7.873720832684077, 8.238719773726350, 8.817740591873026, 9.047549079045157, 9.645476096647777, 10.04967436473114, 10.60131995978503, 11.07328228997050, 11.46630369166040, 11.93540816417196, 12.34996788451842, 13.07748616260286
