| L(s) = 1 | + (0.658 + 1.47i)2-s + (−1.08 + 1.20i)4-s + (−0.406 + 0.913i)5-s + (0.809 − 1.40i)7-s + (−0.951 − 0.309i)8-s − 1.61·10-s + (0.406 + 0.913i)11-s + (2.60 + 0.273i)14-s + (0.951 + 0.309i)17-s + (−0.309 + 0.951i)19-s + (−0.658 − 1.47i)20-s + (−1.08 + 1.20i)22-s + (−0.994 − 0.104i)23-s + (−0.669 − 0.743i)25-s + (0.809 + 2.48i)28-s + (−0.128 + 0.604i)29-s + ⋯ |
| L(s) = 1 | + (0.658 + 1.47i)2-s + (−1.08 + 1.20i)4-s + (−0.406 + 0.913i)5-s + (0.809 − 1.40i)7-s + (−0.951 − 0.309i)8-s − 1.61·10-s + (0.406 + 0.913i)11-s + (2.60 + 0.273i)14-s + (0.951 + 0.309i)17-s + (−0.309 + 0.951i)19-s + (−0.658 − 1.47i)20-s + (−1.08 + 1.20i)22-s + (−0.994 − 0.104i)23-s + (−0.669 − 0.743i)25-s + (0.809 + 2.48i)28-s + (−0.128 + 0.604i)29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.872 - 0.487i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.872 - 0.487i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.617378070\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.617378070\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + (0.406 - 0.913i)T \) |
| good | 2 | \( 1 + (-0.658 - 1.47i)T + (-0.669 + 0.743i)T^{2} \) |
| 7 | \( 1 + (-0.809 + 1.40i)T + (-0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (-0.406 - 0.913i)T + (-0.669 + 0.743i)T^{2} \) |
| 13 | \( 1 + (0.669 + 0.743i)T^{2} \) |
| 17 | \( 1 + (-0.951 - 0.309i)T + (0.809 + 0.587i)T^{2} \) |
| 19 | \( 1 + (0.309 - 0.951i)T + (-0.809 - 0.587i)T^{2} \) |
| 23 | \( 1 + (0.994 + 0.104i)T + (0.978 + 0.207i)T^{2} \) |
| 29 | \( 1 + (0.128 - 0.604i)T + (-0.913 - 0.406i)T^{2} \) |
| 31 | \( 1 + (0.913 - 0.406i)T^{2} \) |
| 37 | \( 1 + (-0.5 + 0.363i)T + (0.309 - 0.951i)T^{2} \) |
| 41 | \( 1 + (-0.406 + 0.913i)T + (-0.669 - 0.743i)T^{2} \) |
| 43 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 47 | \( 1 + (-0.128 + 0.604i)T + (-0.913 - 0.406i)T^{2} \) |
| 53 | \( 1 + (1.53 - 0.5i)T + (0.809 - 0.587i)T^{2} \) |
| 59 | \( 1 + (-0.669 - 0.743i)T^{2} \) |
| 61 | \( 1 + (0.913 - 0.406i)T + (0.669 - 0.743i)T^{2} \) |
| 67 | \( 1 + (-1.58 + 0.336i)T + (0.913 - 0.406i)T^{2} \) |
| 71 | \( 1 + (-0.587 + 0.190i)T + (0.809 - 0.587i)T^{2} \) |
| 73 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 79 | \( 1 + (-0.604 - 0.128i)T + (0.913 + 0.406i)T^{2} \) |
| 83 | \( 1 + (-1.20 + 1.08i)T + (0.104 - 0.994i)T^{2} \) |
| 89 | \( 1 + (-0.587 + 0.809i)T + (-0.309 - 0.951i)T^{2} \) |
| 97 | \( 1 + (0.978 + 0.207i)T + (0.913 + 0.406i)T^{2} \) |
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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
−9.648855662409222627957577674623, −8.290964416426747281968670875001, −7.71724512133288182218206897259, −7.36378058099552671984512556327, −6.60881408818898429370450633879, −5.88163008675857921057849790929, −4.82411197110975556831940292964, −4.05097857367304250921367490676, −3.61535207302980144609769863951, −1.77185956961118158341744492544,
1.05855294438552022672039004866, 2.10726935284601061721497614794, 3.02181317791821850648946541618, 3.98921884107341092260089574970, 4.84499791008365961274383840133, 5.39284091487447534990618598929, 6.21967472153827204496766055983, 7.905877864815911635096239493840, 8.311627170893423319566816436771, 9.366324793069934255621070039832
