L(s) = 1 | + (1.26 − 0.992i)2-s + (0.814 − 0.580i)3-s + (0.136 − 0.563i)4-s + (1.26 + 0.243i)5-s + (0.452 − 1.54i)6-s + (2.62 − 0.329i)7-s + (0.947 + 2.07i)8-s + (0.327 − 0.945i)9-s + (1.83 − 0.946i)10-s + (−1.26 + 1.60i)11-s + (−0.215 − 0.538i)12-s + (−2.39 + 3.72i)13-s + (2.98 − 3.02i)14-s + (1.16 − 0.534i)15-s + (4.28 + 2.21i)16-s + (1.98 − 1.89i)17-s + ⋯ |
L(s) = 1 | + (0.892 − 0.702i)2-s + (0.470 − 0.334i)3-s + (0.0684 − 0.281i)4-s + (0.564 + 0.108i)5-s + (0.184 − 0.629i)6-s + (0.992 − 0.124i)7-s + (0.334 + 0.733i)8-s + (0.109 − 0.315i)9-s + (0.580 − 0.299i)10-s + (−0.379 + 0.483i)11-s + (−0.0622 − 0.155i)12-s + (−0.663 + 1.03i)13-s + (0.798 − 0.807i)14-s + (0.301 − 0.137i)15-s + (1.07 + 0.552i)16-s + (0.482 − 0.459i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.729 + 0.684i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.729 + 0.684i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.68214 - 1.06129i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.68214 - 1.06129i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.814 + 0.580i)T \) |
| 7 | \( 1 + (-2.62 + 0.329i)T \) |
| 23 | \( 1 + (3.24 + 3.53i)T \) |
good | 2 | \( 1 + (-1.26 + 0.992i)T + (0.471 - 1.94i)T^{2} \) |
| 5 | \( 1 + (-1.26 - 0.243i)T + (4.64 + 1.85i)T^{2} \) |
| 11 | \( 1 + (1.26 - 1.60i)T + (-2.59 - 10.6i)T^{2} \) |
| 13 | \( 1 + (2.39 - 3.72i)T + (-5.40 - 11.8i)T^{2} \) |
| 17 | \( 1 + (-1.98 + 1.89i)T + (0.808 - 16.9i)T^{2} \) |
| 19 | \( 1 + (4.78 + 4.56i)T + (0.904 + 18.9i)T^{2} \) |
| 29 | \( 1 + (5.95 + 1.74i)T + (24.3 + 15.6i)T^{2} \) |
| 31 | \( 1 + (-0.335 + 3.51i)T + (-30.4 - 5.86i)T^{2} \) |
| 37 | \( 1 + (-6.98 - 2.41i)T + (29.0 + 22.8i)T^{2} \) |
| 41 | \( 1 + (2.21 + 1.91i)T + (5.83 + 40.5i)T^{2} \) |
| 43 | \( 1 + (-6.36 - 2.90i)T + (28.1 + 32.4i)T^{2} \) |
| 47 | \( 1 + (-2.38 + 1.37i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (10.6 - 0.507i)T + (52.7 - 5.03i)T^{2} \) |
| 59 | \( 1 + (2.79 + 5.41i)T + (-34.2 + 48.0i)T^{2} \) |
| 61 | \( 1 + (2.54 - 3.58i)T + (-19.9 - 57.6i)T^{2} \) |
| 67 | \( 1 + (5.29 - 13.2i)T + (-48.4 - 46.2i)T^{2} \) |
| 71 | \( 1 + (0.00841 + 0.0585i)T + (-68.1 + 20.0i)T^{2} \) |
| 73 | \( 1 + (-12.1 - 2.94i)T + (64.8 + 33.4i)T^{2} \) |
| 79 | \( 1 + (13.2 + 0.633i)T + (78.6 + 7.50i)T^{2} \) |
| 83 | \( 1 + (-0.330 - 0.380i)T + (-11.8 + 82.1i)T^{2} \) |
| 89 | \( 1 + (-9.59 + 0.915i)T + (87.3 - 16.8i)T^{2} \) |
| 97 | \( 1 + (1.63 - 1.88i)T + (-13.8 - 96.0i)T^{2} \) |
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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
−11.19650803566661965854484756090, −10.13729060498460667303005413622, −9.166940329080577577472176939746, −8.070188312239698144577779797079, −7.33184945443248703181563526972, −6.01785707134638876365982823864, −4.75758762216192254814636173107, −4.18047772533869328265831622399, −2.50060702628256651103596003893, −1.98934284546401968743999080563,
1.82340870697435621195157776590, 3.45788715551219048073457005511, 4.54059150604512098516606069007, 5.55090868357551239865005805784, 5.96535710363503461445789376211, 7.63417750144188906893284412637, 8.016337711394258737336381622127, 9.334428596919917581806702742964, 10.22645863178652748249072015738, 10.89873709177587079709725596778