L(s) = 1 | − 1.98·2-s + 2.92·4-s − 0.669·5-s − 3.81·8-s + 9-s + 1.32·10-s + 4.63·16-s + 1.36·17-s − 1.98·18-s − 1.95·20-s − 0.551·25-s − 5.36·32-s − 2.70·34-s + 2.92·36-s − 0.136·37-s + 2.55·40-s + 1.70·43-s − 0.669·45-s − 1.15·47-s + 49-s + 1.09·50-s − 1.15·61-s + 5.99·64-s + 0.406·67-s + 3.99·68-s + 0.920·71-s − 3.81·72-s + ⋯ |
L(s) = 1 | − 1.98·2-s + 2.92·4-s − 0.669·5-s − 3.81·8-s + 9-s + 1.32·10-s + 4.63·16-s + 1.36·17-s − 1.98·18-s − 1.95·20-s − 0.551·25-s − 5.36·32-s − 2.70·34-s + 2.92·36-s − 0.136·37-s + 2.55·40-s + 1.70·43-s − 0.669·45-s − 1.15·47-s + 49-s + 1.09·50-s − 1.15·61-s + 5.99·64-s + 0.406·67-s + 3.99·68-s + 0.920·71-s − 3.81·72-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2671 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2671 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4923753468\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4923753468\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2671 | \( 1+O(T) \) |
good | 2 | \( 1 + 1.98T + T^{2} \) |
| 3 | \( 1 - T^{2} \) |
| 5 | \( 1 + 0.669T + T^{2} \) |
| 7 | \( 1 - T^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 - 1.36T + T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + 0.136T + T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - 1.70T + T^{2} \) |
| 47 | \( 1 + 1.15T + T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 + 1.15T + T^{2} \) |
| 67 | \( 1 - 0.406T + T^{2} \) |
| 71 | \( 1 - 0.920T + T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 + 0.136T + T^{2} \) |
| 89 | \( 1 - 1.70T + T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−9.161944375673616887561806417561, −8.123184996357284135313731387613, −7.75095160594057217829976629555, −7.17711875394691754706932918183, −6.38263635550739633442969813630, −5.47346315765163840545069795751, −3.98534037122319249725510859198, −3.05872573775831576472097522193, −1.89441138228243463250600186449, −0.896812247399687917976850126975,
0.896812247399687917976850126975, 1.89441138228243463250600186449, 3.05872573775831576472097522193, 3.98534037122319249725510859198, 5.47346315765163840545069795751, 6.38263635550739633442969813630, 7.17711875394691754706932918183, 7.75095160594057217829976629555, 8.123184996357284135313731387613, 9.161944375673616887561806417561