L(s) = 1 | + (−0.613 − 1.88i)2-s + (−2.37 + 1.72i)4-s + (0.0388 − 0.119i)5-s + (3.10 + 2.25i)8-s + (0.309 + 0.951i)9-s − 0.249·10-s + (−0.992 + 0.125i)11-s + (0.541 + 1.66i)13-s + (1.44 − 4.46i)16-s + (1.60 − 1.16i)18-s + (1.03 + 0.749i)19-s + (0.113 + 0.350i)20-s + (0.844 + 1.79i)22-s − 1.27·23-s + (0.796 + 0.578i)25-s + (2.81 − 2.04i)26-s + ⋯ |
L(s) = 1 | + (−0.613 − 1.88i)2-s + (−2.37 + 1.72i)4-s + (0.0388 − 0.119i)5-s + (3.10 + 2.25i)8-s + (0.309 + 0.951i)9-s − 0.249·10-s + (−0.992 + 0.125i)11-s + (0.541 + 1.66i)13-s + (1.44 − 4.46i)16-s + (1.60 − 1.16i)18-s + (1.03 + 0.749i)19-s + (0.113 + 0.350i)20-s + (0.844 + 1.79i)22-s − 1.27·23-s + (0.796 + 0.578i)25-s + (2.81 − 2.04i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 869 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.650 + 0.759i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 869 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.650 + 0.759i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5872454501\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5872454501\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 + (0.992 - 0.125i)T \) |
| 79 | \( 1 + (-0.309 - 0.951i)T \) |
good | 2 | \( 1 + (0.613 + 1.88i)T + (-0.809 + 0.587i)T^{2} \) |
| 3 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 5 | \( 1 + (-0.0388 + 0.119i)T + (-0.809 - 0.587i)T^{2} \) |
| 7 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 13 | \( 1 + (-0.541 - 1.66i)T + (-0.809 + 0.587i)T^{2} \) |
| 17 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 19 | \( 1 + (-1.03 - 0.749i)T + (0.309 + 0.951i)T^{2} \) |
| 23 | \( 1 + 1.27T + T^{2} \) |
| 29 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 31 | \( 1 + (0.263 + 0.809i)T + (-0.809 + 0.587i)T^{2} \) |
| 37 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 41 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 53 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 59 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 61 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 67 | \( 1 - 1.07T + T^{2} \) |
| 71 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 73 | \( 1 + (1.56 - 1.13i)T + (0.309 - 0.951i)T^{2} \) |
| 83 | \( 1 + (-0.190 + 0.587i)T + (-0.809 - 0.587i)T^{2} \) |
| 89 | \( 1 - 1.45T + T^{2} \) |
| 97 | \( 1 + (0.574 + 1.76i)T + (-0.809 + 0.587i)T^{2} \) |
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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
−10.21908995136653133963878804532, −9.737775839724417128268652892397, −8.826278037614936079746675361817, −8.055975251092736689435331370270, −7.31817189956906469822512012153, −5.42364628343432899906019979188, −4.49158839911856923318436880281, −3.66027011001660338472885311304, −2.37418355369437312437999438159, −1.59067977021499034058902461662,
0.802447250753115764386888569728, 3.36160395853242410723242515792, 4.70964588012870301912984812928, 5.57628297792713428864445363773, 6.20469313559688313278882268948, 7.16783160333823865056592452183, 7.85868658450927597273696124395, 8.535677120203300277624523530007, 9.387544958759466472878120335315, 10.20145981170883591074356740984