L(s) = 1 | − 2-s + 4-s + 5-s + 7-s − 8-s − 10-s + 6·11-s − 13-s − 14-s + 16-s + 3·17-s − 4·19-s + 20-s − 6·22-s + 3·23-s + 25-s + 26-s + 28-s + 3·29-s + 5·31-s − 32-s − 3·34-s + 35-s − 4·37-s + 4·38-s − 40-s + 6·41-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.447·5-s + 0.377·7-s − 0.353·8-s − 0.316·10-s + 1.80·11-s − 0.277·13-s − 0.267·14-s + 1/4·16-s + 0.727·17-s − 0.917·19-s + 0.223·20-s − 1.27·22-s + 0.625·23-s + 1/5·25-s + 0.196·26-s + 0.188·28-s + 0.557·29-s + 0.898·31-s − 0.176·32-s − 0.514·34-s + 0.169·35-s − 0.657·37-s + 0.648·38-s − 0.158·40-s + 0.937·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1890 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1890 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.632876903\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.632876903\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 7 | \( 1 - T \) |
good | 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 7 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 + 9 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 - 15 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 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
−9.252880838807051595019003742267, −8.540106568759101698195618708428, −7.81049004636027239687421203665, −6.67607852512934235750513553946, −6.42463437316170112258523526721, −5.24267855206183689581964485095, −4.25916995420107621383070297356, −3.18487963889488826424478065824, −1.94673884687353583132457822358, −1.03651296009782685895129399552,
1.03651296009782685895129399552, 1.94673884687353583132457822358, 3.18487963889488826424478065824, 4.25916995420107621383070297356, 5.24267855206183689581964485095, 6.42463437316170112258523526721, 6.67607852512934235750513553946, 7.81049004636027239687421203665, 8.540106568759101698195618708428, 9.252880838807051595019003742267