L(s) = 1 | + (0.907 − 0.419i)2-s + (0.947 + 0.319i)3-s + (0.647 − 0.762i)4-s + (−2.35 − 1.41i)5-s + (0.994 − 0.108i)6-s + (0.260 − 4.80i)7-s + (0.267 − 0.963i)8-s + (0.796 + 0.605i)9-s + (−2.73 − 0.297i)10-s + (−0.0646 + 0.394i)11-s + (0.856 − 0.515i)12-s + (−0.265 + 0.201i)13-s + (−1.77 − 4.46i)14-s + (−1.78 − 2.09i)15-s + (−0.161 − 0.986i)16-s + (0.335 + 6.18i)17-s + ⋯ |
L(s) = 1 | + (0.641 − 0.296i)2-s + (0.547 + 0.184i)3-s + (0.323 − 0.381i)4-s + (−1.05 − 0.634i)5-s + (0.405 − 0.0441i)6-s + (0.0983 − 1.81i)7-s + (0.0945 − 0.340i)8-s + (0.265 + 0.201i)9-s + (−0.864 − 0.0940i)10-s + (−0.0194 + 0.118i)11-s + (0.247 − 0.148i)12-s + (−0.0736 + 0.0559i)13-s + (−0.475 − 1.19i)14-s + (−0.459 − 0.541i)15-s + (−0.0404 − 0.246i)16-s + (0.0812 + 1.49i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.184 + 0.982i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.184 + 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.47818 - 1.22650i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.47818 - 1.22650i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.907 + 0.419i)T \) |
| 3 | \( 1 + (-0.947 - 0.319i)T \) |
| 59 | \( 1 + (5.19 + 5.65i)T \) |
good | 5 | \( 1 + (2.35 + 1.41i)T + (2.34 + 4.41i)T^{2} \) |
| 7 | \( 1 + (-0.260 + 4.80i)T + (-6.95 - 0.756i)T^{2} \) |
| 11 | \( 1 + (0.0646 - 0.394i)T + (-10.4 - 3.51i)T^{2} \) |
| 13 | \( 1 + (0.265 - 0.201i)T + (3.47 - 12.5i)T^{2} \) |
| 17 | \( 1 + (-0.335 - 6.18i)T + (-16.9 + 1.83i)T^{2} \) |
| 19 | \( 1 + (-2.11 + 2.00i)T + (1.02 - 18.9i)T^{2} \) |
| 23 | \( 1 + (-1.98 + 0.437i)T + (20.8 - 9.65i)T^{2} \) |
| 29 | \( 1 + (-7.13 - 3.29i)T + (18.7 + 22.1i)T^{2} \) |
| 31 | \( 1 + (-1.64 - 1.56i)T + (1.67 + 30.9i)T^{2} \) |
| 37 | \( 1 + (1.48 + 5.34i)T + (-31.7 + 19.0i)T^{2} \) |
| 41 | \( 1 + (-7.16 - 1.57i)T + (37.2 + 17.2i)T^{2} \) |
| 43 | \( 1 + (-1.70 - 10.4i)T + (-40.7 + 13.7i)T^{2} \) |
| 47 | \( 1 + (2.53 - 1.52i)T + (22.0 - 41.5i)T^{2} \) |
| 53 | \( 1 + (5.20 - 0.565i)T + (51.7 - 11.3i)T^{2} \) |
| 61 | \( 1 + (6.26 - 2.90i)T + (39.4 - 46.4i)T^{2} \) |
| 67 | \( 1 + (-3.04 + 10.9i)T + (-57.4 - 34.5i)T^{2} \) |
| 71 | \( 1 + (-1.34 + 0.810i)T + (33.2 - 62.7i)T^{2} \) |
| 73 | \( 1 + (-3.49 - 8.77i)T + (-52.9 + 50.2i)T^{2} \) |
| 79 | \( 1 + (-10.7 + 3.60i)T + (62.8 - 47.8i)T^{2} \) |
| 83 | \( 1 + (7.76 - 11.4i)T + (-30.7 - 77.1i)T^{2} \) |
| 89 | \( 1 + (10.8 + 5.02i)T + (57.6 + 67.8i)T^{2} \) |
| 97 | \( 1 + (-1.82 + 4.56i)T + (-70.4 - 66.7i)T^{2} \) |
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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
−11.12539210586688365069503136872, −10.62063605526516758035634594779, −9.558107051993814159550028622752, −8.237848130759296551928679986618, −7.60056312338756515200424216092, −6.55643201499963694430161313147, −4.73721934094526184389248978976, −4.19337135973977192197779534554, −3.27962353841894968944601469810, −1.15259267897184515087792316902,
2.55257899262566018741189111307, 3.26012138923111790595743219888, 4.72747330066285305743666908901, 5.82072556805309254886286585669, 6.96643571517882416603956538749, 7.84937503892500822751024877932, 8.652018897140355169009826382903, 9.649300078650374912121706306697, 11.15180530624812590600986379630, 11.99885519045516182461652010602