L(s) = 1 | − 2-s − 3-s + 4-s + 6-s − 8-s + 9-s − 12-s − 2·13-s + 16-s − 18-s − 2·19-s − 23-s + 24-s − 5·25-s + 2·26-s − 27-s − 6·29-s + 4·31-s − 32-s + 36-s − 10·37-s + 2·38-s + 2·39-s + 6·41-s + 2·43-s + 46-s − 48-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.353·8-s + 1/3·9-s − 0.288·12-s − 0.554·13-s + 1/4·16-s − 0.235·18-s − 0.458·19-s − 0.208·23-s + 0.204·24-s − 25-s + 0.392·26-s − 0.192·27-s − 1.11·29-s + 0.718·31-s − 0.176·32-s + 1/6·36-s − 1.64·37-s + 0.324·38-s + 0.320·39-s + 0.937·41-s + 0.304·43-s + 0.147·46-s − 0.144·48-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6762 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6762 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7368266725\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7368266725\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 23 | \( 1 + T \) |
good | 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 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
−7.931404083423785641892288585422, −7.29332377024181140322458552093, −6.71487145605258108426933912760, −5.87847729116976347092194654844, −5.35805936549294925945573045992, −4.37505316682269263615457205700, −3.61762330685091804253450437465, −2.46892708556937628232716387473, −1.71676922593457090451251489728, −0.49716895631365862715746178734,
0.49716895631365862715746178734, 1.71676922593457090451251489728, 2.46892708556937628232716387473, 3.61762330685091804253450437465, 4.37505316682269263615457205700, 5.35805936549294925945573045992, 5.87847729116976347092194654844, 6.71487145605258108426933912760, 7.29332377024181140322458552093, 7.931404083423785641892288585422