L(s) = 1 | + (−0.187 − 0.982i)2-s + (−0.929 + 0.368i)4-s + (−0.362 + 1.90i)5-s + (0.535 + 0.844i)8-s + (−0.637 + 0.770i)9-s + 1.93·10-s + (0.844 + 0.106i)13-s + (0.728 − 0.684i)16-s + (−0.5 − 0.363i)17-s + (0.876 + 0.481i)18-s + (−0.362 − 1.90i)20-s + (−2.55 − 1.01i)25-s + (−0.0534 − 0.849i)26-s + (1.60 + 0.202i)29-s + (−0.809 − 0.587i)32-s + ⋯ |
L(s) = 1 | + (−0.187 − 0.982i)2-s + (−0.929 + 0.368i)4-s + (−0.362 + 1.90i)5-s + (0.535 + 0.844i)8-s + (−0.637 + 0.770i)9-s + 1.93·10-s + (0.844 + 0.106i)13-s + (0.728 − 0.684i)16-s + (−0.5 − 0.363i)17-s + (0.876 + 0.481i)18-s + (−0.362 − 1.90i)20-s + (−2.55 − 1.01i)25-s + (−0.0534 − 0.849i)26-s + (1.60 + 0.202i)29-s + (−0.809 − 0.587i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 404 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.909 - 0.415i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 404 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.909 - 0.415i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6258329567\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6258329567\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.187 + 0.982i)T \) |
| 101 | \( 1 + (0.929 - 0.368i)T \) |
good | 3 | \( 1 + (0.637 - 0.770i)T^{2} \) |
| 5 | \( 1 + (0.362 - 1.90i)T + (-0.929 - 0.368i)T^{2} \) |
| 7 | \( 1 + (-0.968 + 0.248i)T^{2} \) |
| 11 | \( 1 + (0.187 + 0.982i)T^{2} \) |
| 13 | \( 1 + (-0.844 - 0.106i)T + (0.968 + 0.248i)T^{2} \) |
| 17 | \( 1 + (0.5 + 0.363i)T + (0.309 + 0.951i)T^{2} \) |
| 19 | \( 1 + (-0.0627 - 0.998i)T^{2} \) |
| 23 | \( 1 + (-0.535 + 0.844i)T^{2} \) |
| 29 | \( 1 + (-1.60 - 0.202i)T + (0.968 + 0.248i)T^{2} \) |
| 31 | \( 1 + (-0.968 + 0.248i)T^{2} \) |
| 37 | \( 1 + (-0.791 + 1.68i)T + (-0.637 - 0.770i)T^{2} \) |
| 41 | \( 1 + (1.17 - 0.856i)T + (0.309 - 0.951i)T^{2} \) |
| 43 | \( 1 + (0.992 - 0.125i)T^{2} \) |
| 47 | \( 1 + (0.992 + 0.125i)T^{2} \) |
| 53 | \( 1 + (-1.84 - 0.730i)T + (0.728 + 0.684i)T^{2} \) |
| 59 | \( 1 + (-0.0627 + 0.998i)T^{2} \) |
| 61 | \( 1 + (0.996 - 0.394i)T + (0.728 - 0.684i)T^{2} \) |
| 67 | \( 1 + (0.637 + 0.770i)T^{2} \) |
| 71 | \( 1 + (0.637 - 0.770i)T^{2} \) |
| 73 | \( 1 + (-0.110 + 0.0604i)T + (0.535 - 0.844i)T^{2} \) |
| 79 | \( 1 + (-0.535 - 0.844i)T^{2} \) |
| 83 | \( 1 + (-0.535 - 0.844i)T^{2} \) |
| 89 | \( 1 + (0.929 + 0.872i)T + (0.0627 + 0.998i)T^{2} \) |
| 97 | \( 1 + (-1.84 + 0.730i)T + (0.728 - 0.684i)T^{2} \) |
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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
−11.33721207839740917299584355515, −10.71154316668553788949698142542, −10.19530500832557369822779560473, −8.876002485851172333833648863643, −7.954286293541852403771852648961, −6.99832023618838197538436719024, −5.81921574722615473244405486055, −4.27311624138790197804338693670, −3.14092994858464580420160289037, −2.35680192243366114305958496144,
1.00930115635347560036163001700, 3.81852124290055734276165534742, 4.74925681672323304358952974105, 5.68722531415882950981399204060, 6.59647754748721000216870305386, 8.076894771306308114158974366662, 8.586200294402369596363571788650, 9.119284114263838995142902563221, 10.19460872653083352560989903694, 11.67632513723280884608813024602