L(s) = 1 | + (0.153 + 1.07i)3-s + (0.959 + 0.281i)5-s + (−0.989 + 1.14i)7-s + (−0.162 + 0.0476i)9-s + (−0.153 + 1.07i)15-s + (−1.37 − 0.883i)21-s + (0.540 − 0.841i)23-s + (0.841 + 0.540i)25-s + (0.373 + 0.817i)27-s + (−0.797 + 1.74i)29-s + (−1.27 + 0.817i)35-s + (−1.25 − 0.368i)41-s + (−0.258 − 1.80i)43-s − 0.169·45-s − 0.563·47-s + ⋯ |
L(s) = 1 | + (0.153 + 1.07i)3-s + (0.959 + 0.281i)5-s + (−0.989 + 1.14i)7-s + (−0.162 + 0.0476i)9-s + (−0.153 + 1.07i)15-s + (−1.37 − 0.883i)21-s + (0.540 − 0.841i)23-s + (0.841 + 0.540i)25-s + (0.373 + 0.817i)27-s + (−0.797 + 1.74i)29-s + (−1.27 + 0.817i)35-s + (−1.25 − 0.368i)41-s + (−0.258 − 1.80i)43-s − 0.169·45-s − 0.563·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.320 - 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.320 - 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.277295855\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.277295855\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-0.959 - 0.281i)T \) |
| 23 | \( 1 + (-0.540 + 0.841i)T \) |
good | 3 | \( 1 + (-0.153 - 1.07i)T + (-0.959 + 0.281i)T^{2} \) |
| 7 | \( 1 + (0.989 - 1.14i)T + (-0.142 - 0.989i)T^{2} \) |
| 11 | \( 1 + (-0.415 + 0.909i)T^{2} \) |
| 13 | \( 1 + (0.142 - 0.989i)T^{2} \) |
| 17 | \( 1 + (0.654 + 0.755i)T^{2} \) |
| 19 | \( 1 + (0.654 - 0.755i)T^{2} \) |
| 29 | \( 1 + (0.797 - 1.74i)T + (-0.654 - 0.755i)T^{2} \) |
| 31 | \( 1 + (0.959 + 0.281i)T^{2} \) |
| 37 | \( 1 + (-0.841 + 0.540i)T^{2} \) |
| 41 | \( 1 + (1.25 + 0.368i)T + (0.841 + 0.540i)T^{2} \) |
| 43 | \( 1 + (0.258 + 1.80i)T + (-0.959 + 0.281i)T^{2} \) |
| 47 | \( 1 + 0.563T + T^{2} \) |
| 53 | \( 1 + (0.142 + 0.989i)T^{2} \) |
| 59 | \( 1 + (0.142 - 0.989i)T^{2} \) |
| 61 | \( 1 + (-0.118 + 0.822i)T + (-0.959 - 0.281i)T^{2} \) |
| 67 | \( 1 + (-1.66 - 1.07i)T + (0.415 + 0.909i)T^{2} \) |
| 71 | \( 1 + (-0.415 - 0.909i)T^{2} \) |
| 73 | \( 1 + (0.654 - 0.755i)T^{2} \) |
| 79 | \( 1 + (0.142 - 0.989i)T^{2} \) |
| 83 | \( 1 + (-1.89 + 0.557i)T + (0.841 - 0.540i)T^{2} \) |
| 89 | \( 1 + (0.239 + 1.66i)T + (-0.959 + 0.281i)T^{2} \) |
| 97 | \( 1 + (-0.841 - 0.540i)T^{2} \) |
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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
−9.619443407228338329502943660093, −9.041504028497365764953209481815, −8.610184758386035532441791083161, −7.02187270974985577459881712585, −6.49544175421944129493320677283, −5.44624260034562421010173223222, −5.04207131442443271005814700038, −3.65063675794668156520425247782, −3.01056238020387829426521311514, −1.96121763690997354116373225554,
0.971254721685070402317675617533, 1.98918162592237202362650938239, 3.10525283938713685138350798467, 4.18133603434797674723010347522, 5.29478984869608291444356212624, 6.39151092401094530674636859296, 6.66092022582596609068155523984, 7.58625336509898968117259837596, 8.202129347206729552792592040463, 9.510050811679417166504635695534