L(s) = 1 | + 2-s − 3-s + 4-s + 2·5-s − 6-s − 2·7-s + 8-s + 9-s + 2·10-s + 11-s − 12-s + 4·13-s − 2·14-s − 2·15-s + 16-s + 17-s + 18-s − 2·19-s + 2·20-s + 2·21-s + 22-s + 2·23-s − 24-s − 25-s + 4·26-s − 27-s − 2·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.894·5-s − 0.408·6-s − 0.755·7-s + 0.353·8-s + 1/3·9-s + 0.632·10-s + 0.301·11-s − 0.288·12-s + 1.10·13-s − 0.534·14-s − 0.516·15-s + 1/4·16-s + 0.242·17-s + 0.235·18-s − 0.458·19-s + 0.447·20-s + 0.436·21-s + 0.213·22-s + 0.417·23-s − 0.204·24-s − 1/5·25-s + 0.784·26-s − 0.192·27-s − 0.377·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1122 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1122 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.456721278\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.456721278\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 17 | \( 1 - T \) |
good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 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
−10.06578207306856404840701986875, −9.150940393068328601866793933182, −8.169804735948644622486108184404, −6.84940851749676675991664303351, −6.29457653019056207189965811391, −5.75603308684022661568940417710, −4.70874545888518294082453186137, −3.70209117565062812459761521428, −2.60060005311032819962750372622, −1.21513589446268367082695910471,
1.21513589446268367082695910471, 2.60060005311032819962750372622, 3.70209117565062812459761521428, 4.70874545888518294082453186137, 5.75603308684022661568940417710, 6.29457653019056207189965811391, 6.84940851749676675991664303351, 8.169804735948644622486108184404, 9.150940393068328601866793933182, 10.06578207306856404840701986875