L(s) = 1 | + (1.69 + 0.933i)3-s + (0.728 − 0.684i)4-s + (−0.809 + 0.587i)5-s + (1.47 + 2.32i)9-s + (−0.929 + 0.368i)11-s + (1.87 − 0.481i)12-s + (−1.92 + 0.242i)15-s + (0.0627 − 0.998i)16-s + (−0.187 + 0.982i)20-s + (−1.11 − 1.35i)23-s + (0.309 − 0.951i)25-s + (0.213 + 3.38i)27-s + (−0.929 − 0.872i)31-s + (−1.92 − 0.242i)33-s + (2.66 + 0.684i)36-s + (0.0388 − 0.616i)37-s + ⋯ |
L(s) = 1 | + (1.69 + 0.933i)3-s + (0.728 − 0.684i)4-s + (−0.809 + 0.587i)5-s + (1.47 + 2.32i)9-s + (−0.929 + 0.368i)11-s + (1.87 − 0.481i)12-s + (−1.92 + 0.242i)15-s + (0.0627 − 0.998i)16-s + (−0.187 + 0.982i)20-s + (−1.11 − 1.35i)23-s + (0.309 − 0.951i)25-s + (0.213 + 3.38i)27-s + (−0.929 − 0.872i)31-s + (−1.92 − 0.242i)33-s + (2.66 + 0.684i)36-s + (0.0388 − 0.616i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.675 - 0.737i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.675 - 0.737i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.845585364\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.845585364\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (0.809 - 0.587i)T \) |
| 11 | \( 1 + (0.929 - 0.368i)T \) |
good | 2 | \( 1 + (-0.728 + 0.684i)T^{2} \) |
| 3 | \( 1 + (-1.69 - 0.933i)T + (0.535 + 0.844i)T^{2} \) |
| 7 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 13 | \( 1 + (0.425 - 0.904i)T^{2} \) |
| 17 | \( 1 + (-0.0627 - 0.998i)T^{2} \) |
| 19 | \( 1 + (-0.535 + 0.844i)T^{2} \) |
| 23 | \( 1 + (1.11 + 1.35i)T + (-0.187 + 0.982i)T^{2} \) |
| 29 | \( 1 + (0.929 + 0.368i)T^{2} \) |
| 31 | \( 1 + (0.929 + 0.872i)T + (0.0627 + 0.998i)T^{2} \) |
| 37 | \( 1 + (-0.0388 + 0.616i)T + (-0.992 - 0.125i)T^{2} \) |
| 41 | \( 1 + (0.187 + 0.982i)T^{2} \) |
| 43 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 47 | \( 1 + (-0.688 - 1.46i)T + (-0.637 + 0.770i)T^{2} \) |
| 53 | \( 1 + (-1.84 + 0.233i)T + (0.968 - 0.248i)T^{2} \) |
| 59 | \( 1 + (-0.121 + 0.0312i)T + (0.876 - 0.481i)T^{2} \) |
| 61 | \( 1 + (0.187 - 0.982i)T^{2} \) |
| 67 | \( 1 + (-0.371 - 1.94i)T + (-0.929 + 0.368i)T^{2} \) |
| 71 | \( 1 + (0.824 + 1.75i)T + (-0.637 + 0.770i)T^{2} \) |
| 73 | \( 1 + (-0.876 - 0.481i)T^{2} \) |
| 79 | \( 1 + (-0.535 - 0.844i)T^{2} \) |
| 83 | \( 1 + (-0.535 + 0.844i)T^{2} \) |
| 89 | \( 1 + (0.824 + 0.211i)T + (0.876 + 0.481i)T^{2} \) |
| 97 | \( 1 + (0.0235 - 0.123i)T + (-0.929 - 0.368i)T^{2} \) |
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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
−10.04384694787498792095521196313, −9.115030073621483619365418849495, −8.211264862607639919259981540636, −7.62287439335752797907717134421, −6.97041086790168344317646960087, −5.63254448340369198504372639709, −4.50532923917120063409263396086, −3.79534107220153769052820767152, −2.65480237646948388246830923936, −2.23674837432607368230048325067,
1.54599592651136661385259086604, 2.57715608706364383673357699118, 3.43335697259508882133500371682, 4.03341771062005035857103628121, 5.64632090242308363295709046175, 6.96823233481356997596464170985, 7.42230315773651402544439701681, 8.098740681533786213485673383107, 8.498172069863132264528238509030, 9.291678960941409430635860891825