| L(s) = 1 | + (−0.249 + 0.287i)3-s + (0.142 − 0.989i)5-s + (0.948 + 2.07i)7-s + (0.406 + 2.82i)9-s + (−1.15 + 0.338i)11-s + (−0.598 + 1.31i)13-s + (0.249 + 0.287i)15-s + (2.71 − 1.74i)17-s + (3.45 + 2.21i)19-s + (−0.834 − 0.244i)21-s + (0.285 + 4.78i)23-s + (−0.959 − 0.281i)25-s + (−1.87 − 1.20i)27-s + (7.48 − 4.80i)29-s + (3.34 + 3.85i)31-s + ⋯ |
| L(s) = 1 | + (−0.144 + 0.166i)3-s + (0.0636 − 0.442i)5-s + (0.358 + 0.784i)7-s + (0.135 + 0.941i)9-s + (−0.347 + 0.101i)11-s + (−0.166 + 0.363i)13-s + (0.0644 + 0.0743i)15-s + (0.659 − 0.423i)17-s + (0.791 + 0.508i)19-s + (−0.182 − 0.0534i)21-s + (0.0596 + 0.998i)23-s + (−0.191 − 0.0563i)25-s + (−0.361 − 0.232i)27-s + (1.38 − 0.892i)29-s + (0.600 + 0.693i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.590 - 0.807i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.590 - 0.807i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.21670 + 0.617669i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.21670 + 0.617669i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (-0.285 - 4.78i)T \) |
| good | 3 | \( 1 + (0.249 - 0.287i)T + (-0.426 - 2.96i)T^{2} \) |
| 7 | \( 1 + (-0.948 - 2.07i)T + (-4.58 + 5.29i)T^{2} \) |
| 11 | \( 1 + (1.15 - 0.338i)T + (9.25 - 5.94i)T^{2} \) |
| 13 | \( 1 + (0.598 - 1.31i)T + (-8.51 - 9.82i)T^{2} \) |
| 17 | \( 1 + (-2.71 + 1.74i)T + (7.06 - 15.4i)T^{2} \) |
| 19 | \( 1 + (-3.45 - 2.21i)T + (7.89 + 17.2i)T^{2} \) |
| 29 | \( 1 + (-7.48 + 4.80i)T + (12.0 - 26.3i)T^{2} \) |
| 31 | \( 1 + (-3.34 - 3.85i)T + (-4.41 + 30.6i)T^{2} \) |
| 37 | \( 1 + (-0.528 - 3.67i)T + (-35.5 + 10.4i)T^{2} \) |
| 41 | \( 1 + (0.982 - 6.83i)T + (-39.3 - 11.5i)T^{2} \) |
| 43 | \( 1 + (6.09 - 7.03i)T + (-6.11 - 42.5i)T^{2} \) |
| 47 | \( 1 + 5.96T + 47T^{2} \) |
| 53 | \( 1 + (5.25 + 11.5i)T + (-34.7 + 40.0i)T^{2} \) |
| 59 | \( 1 + (-2.77 + 6.08i)T + (-38.6 - 44.5i)T^{2} \) |
| 61 | \( 1 + (8.39 + 9.68i)T + (-8.68 + 60.3i)T^{2} \) |
| 67 | \( 1 + (-6.69 - 1.96i)T + (56.3 + 36.2i)T^{2} \) |
| 71 | \( 1 + (-7.66 - 2.24i)T + (59.7 + 38.3i)T^{2} \) |
| 73 | \( 1 + (3.23 + 2.08i)T + (30.3 + 66.4i)T^{2} \) |
| 79 | \( 1 + (-3.99 + 8.74i)T + (-51.7 - 59.7i)T^{2} \) |
| 83 | \( 1 + (-0.0525 - 0.365i)T + (-79.6 + 23.3i)T^{2} \) |
| 89 | \( 1 + (-6.81 + 7.86i)T + (-12.6 - 88.0i)T^{2} \) |
| 97 | \( 1 + (-1.48 + 10.3i)T + (-93.0 - 27.3i)T^{2} \) |
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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
−11.44766365346568395265652464998, −10.07650483401695760448726332267, −9.617153891903391410068837381837, −8.269639233309992285218250999284, −7.85035437080796561963725954053, −6.44001481024870845166780445463, −5.21034549602435459728908227630, −4.80745673198029298306034549173, −3.08744065430744563371883234096, −1.68624781036009272551634830239,
0.956988850756816685261629953841, 2.85400791419741965671112691185, 3.98724733443639050267503598757, 5.22632109125693620437681037091, 6.39206472407179085682378425976, 7.18627925207510417612119444935, 8.068364941557681572645666097835, 9.181121578716794395447847739997, 10.27010310589876700988400274281, 10.72331029014439299211522745613
