| L(s) = 1 | + (0.128 − 0.604i)2-s + (0.564 + 0.251i)4-s + (0.207 + 0.978i)5-s + (−0.309 + 0.535i)7-s + (0.587 − 0.809i)8-s + 0.618·10-s + (−0.207 + 0.978i)11-s + (0.283 + 0.255i)14-s + (−0.587 + 0.809i)17-s + (0.809 + 0.587i)19-s + (−0.128 + 0.604i)20-s + (0.564 + 0.251i)22-s + (−0.743 − 0.669i)23-s + (−0.913 + 0.406i)25-s + (−0.309 + 0.224i)28-s + (−1.60 + 0.169i)29-s + ⋯ |
| L(s) = 1 | + (0.128 − 0.604i)2-s + (0.564 + 0.251i)4-s + (0.207 + 0.978i)5-s + (−0.309 + 0.535i)7-s + (0.587 − 0.809i)8-s + 0.618·10-s + (−0.207 + 0.978i)11-s + (0.283 + 0.255i)14-s + (−0.587 + 0.809i)17-s + (0.809 + 0.587i)19-s + (−0.128 + 0.604i)20-s + (0.564 + 0.251i)22-s + (−0.743 − 0.669i)23-s + (−0.913 + 0.406i)25-s + (−0.309 + 0.224i)28-s + (−1.60 + 0.169i)29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.813 - 0.582i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.813 - 0.582i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.444104013\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.444104013\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + (-0.207 - 0.978i)T \) |
| good | 2 | \( 1 + (-0.128 + 0.604i)T + (-0.913 - 0.406i)T^{2} \) |
| 7 | \( 1 + (0.309 - 0.535i)T + (-0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (0.207 - 0.978i)T + (-0.913 - 0.406i)T^{2} \) |
| 13 | \( 1 + (0.913 - 0.406i)T^{2} \) |
| 17 | \( 1 + (0.587 - 0.809i)T + (-0.309 - 0.951i)T^{2} \) |
| 19 | \( 1 + (-0.809 - 0.587i)T + (0.309 + 0.951i)T^{2} \) |
| 23 | \( 1 + (0.743 + 0.669i)T + (0.104 + 0.994i)T^{2} \) |
| 29 | \( 1 + (1.60 - 0.169i)T + (0.978 - 0.207i)T^{2} \) |
| 31 | \( 1 + (-0.978 - 0.207i)T^{2} \) |
| 37 | \( 1 + (-0.5 + 1.53i)T + (-0.809 - 0.587i)T^{2} \) |
| 41 | \( 1 + (0.207 + 0.978i)T + (-0.913 + 0.406i)T^{2} \) |
| 43 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 47 | \( 1 + (-1.60 + 0.169i)T + (0.978 - 0.207i)T^{2} \) |
| 53 | \( 1 + (0.363 + 0.5i)T + (-0.309 + 0.951i)T^{2} \) |
| 59 | \( 1 + (-0.913 + 0.406i)T^{2} \) |
| 61 | \( 1 + (-0.978 - 0.207i)T + (0.913 + 0.406i)T^{2} \) |
| 67 | \( 1 + (0.0646 - 0.614i)T + (-0.978 - 0.207i)T^{2} \) |
| 71 | \( 1 + (-0.951 - 1.30i)T + (-0.309 + 0.951i)T^{2} \) |
| 73 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 79 | \( 1 + (0.169 + 1.60i)T + (-0.978 + 0.207i)T^{2} \) |
| 83 | \( 1 + (0.251 + 0.564i)T + (-0.669 + 0.743i)T^{2} \) |
| 89 | \( 1 + (-0.951 + 0.309i)T + (0.809 - 0.587i)T^{2} \) |
| 97 | \( 1 + (0.104 + 0.994i)T + (-0.978 + 0.207i)T^{2} \) |
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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
−9.708494846370974699249086395731, −8.740942328905492445174216192464, −7.47088402309264215479694959267, −7.28950927811480562641446349255, −6.21852587913155190686570442271, −5.63453626208939596600982241399, −4.14274902895136587242959485164, −3.52504919691439498338183630100, −2.36371131692788573142729831514, −1.98348722708480814376015696402,
0.999774034346776602776401196314, 2.29425172759656909618120534830, 3.48839457384763639183199104890, 4.64480222922160333758692894443, 5.40109728250248379231182444991, 6.01750613479622025642176961628, 6.89151968948890159587595934836, 7.67306388002306968150565671290, 8.284864856520159676346984219673, 9.296740203023849365874080254182
