| L(s) = 1 | + (0.173 − 0.984i)3-s + (1.30 + 1.09i)5-s + (−1.57 + 0.909i)7-s + (−0.939 − 0.342i)9-s + (−5.51 − 3.18i)11-s + (−2.14 + 0.378i)13-s + (1.30 − 1.09i)15-s + (−4.03 + 1.46i)17-s + (−4.17 + 1.23i)19-s + (0.622 + 1.70i)21-s + (−0.257 − 0.307i)23-s + (−0.368 − 2.08i)25-s + (−0.5 + 0.866i)27-s + (1.76 − 4.85i)29-s + (−2.11 − 3.66i)31-s + ⋯ |
| L(s) = 1 | + (0.100 − 0.568i)3-s + (0.581 + 0.487i)5-s + (−0.595 + 0.343i)7-s + (−0.313 − 0.114i)9-s + (−1.66 − 0.960i)11-s + (−0.596 + 0.105i)13-s + (0.335 − 0.281i)15-s + (−0.977 + 0.355i)17-s + (−0.958 + 0.283i)19-s + (0.135 + 0.373i)21-s + (−0.0537 − 0.0640i)23-s + (−0.0736 − 0.417i)25-s + (−0.0962 + 0.166i)27-s + (0.327 − 0.900i)29-s + (−0.380 − 0.658i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0590i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.998 + 0.0590i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.00736015 - 0.249046i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.00736015 - 0.249046i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.173 + 0.984i)T \) |
| 19 | \( 1 + (4.17 - 1.23i)T \) |
| good | 5 | \( 1 + (-1.30 - 1.09i)T + (0.868 + 4.92i)T^{2} \) |
| 7 | \( 1 + (1.57 - 0.909i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (5.51 + 3.18i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (2.14 - 0.378i)T + (12.2 - 4.44i)T^{2} \) |
| 17 | \( 1 + (4.03 - 1.46i)T + (13.0 - 10.9i)T^{2} \) |
| 23 | \( 1 + (0.257 + 0.307i)T + (-3.99 + 22.6i)T^{2} \) |
| 29 | \( 1 + (-1.76 + 4.85i)T + (-22.2 - 18.6i)T^{2} \) |
| 31 | \( 1 + (2.11 + 3.66i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 0.939iT - 37T^{2} \) |
| 41 | \( 1 + (-7.52 - 1.32i)T + (38.5 + 14.0i)T^{2} \) |
| 43 | \( 1 + (-3.71 + 4.42i)T + (-7.46 - 42.3i)T^{2} \) |
| 47 | \( 1 + (-3.21 + 8.83i)T + (-36.0 - 30.2i)T^{2} \) |
| 53 | \( 1 + (-0.589 - 0.702i)T + (-9.20 + 52.1i)T^{2} \) |
| 59 | \( 1 + (10.8 - 3.95i)T + (45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (10.5 - 8.87i)T + (10.5 - 60.0i)T^{2} \) |
| 67 | \( 1 + (-6.48 - 2.36i)T + (51.3 + 43.0i)T^{2} \) |
| 71 | \( 1 + (-0.701 - 0.588i)T + (12.3 + 69.9i)T^{2} \) |
| 73 | \( 1 + (2.33 - 13.2i)T + (-68.5 - 24.9i)T^{2} \) |
| 79 | \( 1 + (2.36 - 13.3i)T + (-74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (-1.52 + 0.879i)T + (41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (2.20 - 0.388i)T + (83.6 - 30.4i)T^{2} \) |
| 97 | \( 1 + (1.26 + 3.46i)T + (-74.3 + 62.3i)T^{2} \) |
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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
−9.754961479805963919946933732052, −8.736936395354006546534368536908, −8.027519454375422003087172201189, −7.08469175423362366025783731632, −6.09019827043688210943348345517, −5.69623157845468757292796309352, −4.24656410883922890670778007678, −2.71266430107057197522716840317, −2.32827049703483158577367028842, −0.10249267613275242517256649654,
2.09891432057710813836247402853, 3.06818609755959274515588671383, 4.59131049318031372814018949287, 4.97816143563083589475727223471, 6.11926819936637677707368089705, 7.17415463089632802320886658897, 7.979394558450588448598902692850, 9.190993414321650869064160442653, 9.516147800160530781007780000487, 10.60720140693592622741155515355
