L(s) = 1 | + (0.542 − 0.656i)3-s + (0.968 − 0.248i)4-s + (0.309 − 0.951i)5-s + (0.0515 + 0.269i)9-s + (−0.992 + 0.125i)11-s + (0.362 − 0.770i)12-s + (−0.456 − 0.718i)15-s + (0.876 − 0.481i)16-s + (0.0627 − 0.998i)20-s + (−0.929 + 0.872i)23-s + (−0.809 − 0.587i)25-s + (0.951 + 0.522i)27-s + (1.41 + 0.362i)31-s + (−0.456 + 0.718i)33-s + (0.117 + 0.248i)36-s + (−1.41 + 0.779i)37-s + ⋯ |
L(s) = 1 | + (0.542 − 0.656i)3-s + (0.968 − 0.248i)4-s + (0.309 − 0.951i)5-s + (0.0515 + 0.269i)9-s + (−0.992 + 0.125i)11-s + (0.362 − 0.770i)12-s + (−0.456 − 0.718i)15-s + (0.876 − 0.481i)16-s + (0.0627 − 0.998i)20-s + (−0.929 + 0.872i)23-s + (−0.809 − 0.587i)25-s + (0.951 + 0.522i)27-s + (1.41 + 0.362i)31-s + (−0.456 + 0.718i)33-s + (0.117 + 0.248i)36-s + (−1.41 + 0.779i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.448 + 0.893i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.448 + 0.893i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.603342230\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.603342230\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (-0.309 + 0.951i)T \) |
| 11 | \( 1 + (0.992 - 0.125i)T \) |
good | 2 | \( 1 + (-0.968 + 0.248i)T^{2} \) |
| 3 | \( 1 + (-0.542 + 0.656i)T + (-0.187 - 0.982i)T^{2} \) |
| 7 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 13 | \( 1 + (0.929 - 0.368i)T^{2} \) |
| 17 | \( 1 + (-0.876 - 0.481i)T^{2} \) |
| 19 | \( 1 + (0.187 - 0.982i)T^{2} \) |
| 23 | \( 1 + (0.929 - 0.872i)T + (0.0627 - 0.998i)T^{2} \) |
| 29 | \( 1 + (0.992 + 0.125i)T^{2} \) |
| 31 | \( 1 + (-1.41 - 0.362i)T + (0.876 + 0.481i)T^{2} \) |
| 37 | \( 1 + (1.41 - 0.779i)T + (0.535 - 0.844i)T^{2} \) |
| 41 | \( 1 + (-0.0627 - 0.998i)T^{2} \) |
| 43 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 47 | \( 1 + (0.574 + 0.227i)T + (0.728 + 0.684i)T^{2} \) |
| 53 | \( 1 + (1.06 + 1.67i)T + (-0.425 + 0.904i)T^{2} \) |
| 59 | \( 1 + (0.746 - 1.58i)T + (-0.637 - 0.770i)T^{2} \) |
| 61 | \( 1 + (-0.0627 + 0.998i)T^{2} \) |
| 67 | \( 1 + (-0.0672 - 1.06i)T + (-0.992 + 0.125i)T^{2} \) |
| 71 | \( 1 + (-0.791 - 0.313i)T + (0.728 + 0.684i)T^{2} \) |
| 73 | \( 1 + (0.637 - 0.770i)T^{2} \) |
| 79 | \( 1 + (0.187 + 0.982i)T^{2} \) |
| 83 | \( 1 + (0.187 - 0.982i)T^{2} \) |
| 89 | \( 1 + (-0.791 - 1.68i)T + (-0.637 + 0.770i)T^{2} \) |
| 97 | \( 1 + (-0.110 + 1.74i)T + (-0.992 - 0.125i)T^{2} \) |
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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
−9.844233907951156048918128983272, −8.483494219007346031422584877843, −8.098090027220845182478729616195, −7.30547454218038929498186547952, −6.44736008200831344076734421103, −5.47605129826778314227917329444, −4.79083622601090308267121274572, −3.23068229612603698777377824179, −2.20348861378972512932701772302, −1.47188784177027240342647154405,
2.08619148406347412890827850148, 2.92569531630095994240828212074, 3.56278989398208822373116171711, 4.77618180725886782506625995222, 6.11964567214697439021722333394, 6.50671015167695544107503843952, 7.61933754642618624548824302283, 8.145538905671528860038289730174, 9.246021042619522304673141000510, 10.14687929349737243972271219545