| L(s) = 1 | + (−1.26 + 1.46i)3-s + (−0.142 + 0.989i)5-s + (0.0191 + 0.0418i)7-s + (−0.107 − 0.746i)9-s + (0.229 − 0.0674i)11-s + (−1.75 + 3.83i)13-s + (−1.26 − 1.46i)15-s + (−0.597 + 0.383i)17-s + (−3.34 − 2.14i)19-s + (−0.0856 − 0.0251i)21-s + (−4.35 + 2.00i)23-s + (−0.959 − 0.281i)25-s + (−3.66 − 2.35i)27-s + (−0.590 + 0.379i)29-s + (−3.43 − 3.96i)31-s + ⋯ |
| L(s) = 1 | + (−0.732 + 0.845i)3-s + (−0.0636 + 0.442i)5-s + (0.00723 + 0.0158i)7-s + (−0.0357 − 0.248i)9-s + (0.0692 − 0.0203i)11-s + (−0.485 + 1.06i)13-s + (−0.327 − 0.378i)15-s + (−0.144 + 0.0930i)17-s + (−0.766 − 0.492i)19-s + (−0.0186 − 0.00548i)21-s + (−0.908 + 0.417i)23-s + (−0.191 − 0.0563i)25-s + (−0.704 − 0.452i)27-s + (−0.109 + 0.0704i)29-s + (−0.617 − 0.712i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.966 - 0.257i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.966 - 0.257i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0825898 + 0.631637i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0825898 + 0.631637i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (0.142 - 0.989i)T \) |
| 23 | \( 1 + (4.35 - 2.00i)T \) |
| good | 3 | \( 1 + (1.26 - 1.46i)T + (-0.426 - 2.96i)T^{2} \) |
| 7 | \( 1 + (-0.0191 - 0.0418i)T + (-4.58 + 5.29i)T^{2} \) |
| 11 | \( 1 + (-0.229 + 0.0674i)T + (9.25 - 5.94i)T^{2} \) |
| 13 | \( 1 + (1.75 - 3.83i)T + (-8.51 - 9.82i)T^{2} \) |
| 17 | \( 1 + (0.597 - 0.383i)T + (7.06 - 15.4i)T^{2} \) |
| 19 | \( 1 + (3.34 + 2.14i)T + (7.89 + 17.2i)T^{2} \) |
| 29 | \( 1 + (0.590 - 0.379i)T + (12.0 - 26.3i)T^{2} \) |
| 31 | \( 1 + (3.43 + 3.96i)T + (-4.41 + 30.6i)T^{2} \) |
| 37 | \( 1 + (-0.347 - 2.41i)T + (-35.5 + 10.4i)T^{2} \) |
| 41 | \( 1 + (0.510 - 3.55i)T + (-39.3 - 11.5i)T^{2} \) |
| 43 | \( 1 + (3.44 - 3.97i)T + (-6.11 - 42.5i)T^{2} \) |
| 47 | \( 1 + 0.0114T + 47T^{2} \) |
| 53 | \( 1 + (-5.16 - 11.3i)T + (-34.7 + 40.0i)T^{2} \) |
| 59 | \( 1 + (2.03 - 4.45i)T + (-38.6 - 44.5i)T^{2} \) |
| 61 | \( 1 + (-3.47 - 4.01i)T + (-8.68 + 60.3i)T^{2} \) |
| 67 | \( 1 + (-2.04 - 0.599i)T + (56.3 + 36.2i)T^{2} \) |
| 71 | \( 1 + (-7.02 - 2.06i)T + (59.7 + 38.3i)T^{2} \) |
| 73 | \( 1 + (0.786 + 0.505i)T + (30.3 + 66.4i)T^{2} \) |
| 79 | \( 1 + (-4.12 + 9.03i)T + (-51.7 - 59.7i)T^{2} \) |
| 83 | \( 1 + (-1.46 - 10.2i)T + (-79.6 + 23.3i)T^{2} \) |
| 89 | \( 1 + (-1.17 + 1.35i)T + (-12.6 - 88.0i)T^{2} \) |
| 97 | \( 1 + (0.0699 - 0.486i)T + (-93.0 - 27.3i)T^{2} \) |
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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
−11.40575176665329613741100054402, −10.59197201129091028894648462664, −9.853660816903410129541684681479, −9.008882638800262496229920822882, −7.75736653741611046926933915025, −6.70714294575405661030302858859, −5.78592589036053907526075944606, −4.67026591706311312434893453074, −3.89863100400416437440774351156, −2.22822279154931786444936253954,
0.41366832324406608780235736168, 1.99982120561146105843800862598, 3.73423905169882131674906695515, 5.12040050101071654329693708306, 5.94689794384113072062857014576, 6.89040471367440605842792709241, 7.81065015082353765915687213438, 8.690681978633750363328901846673, 9.877638039813394808075596846021, 10.74140188695569822175599675479
