L(s) = 1 | + (0.781 − 0.623i)2-s + (0.222 − 0.974i)4-s + (−0.433 − 0.900i)8-s + (−0.988 + 0.149i)11-s + (−0.149 − 0.988i)13-s + (−0.900 − 0.433i)16-s + (0.680 + 0.733i)17-s + (0.5 − 0.866i)19-s + (−0.680 + 0.733i)22-s + (0.294 + 0.955i)23-s + (−0.733 − 0.680i)26-s + (0.733 − 0.680i)29-s + 31-s + (−0.974 + 0.222i)32-s + (0.988 + 0.149i)34-s + ⋯ |
L(s) = 1 | + (0.781 − 0.623i)2-s + (0.222 − 0.974i)4-s + (−0.433 − 0.900i)8-s + (−0.988 + 0.149i)11-s + (−0.149 − 0.988i)13-s + (−0.900 − 0.433i)16-s + (0.680 + 0.733i)17-s + (0.5 − 0.866i)19-s + (−0.680 + 0.733i)22-s + (0.294 + 0.955i)23-s + (−0.733 − 0.680i)26-s + (0.733 − 0.680i)29-s + 31-s + (−0.974 + 0.222i)32-s + (0.988 + 0.149i)34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.958 + 0.285i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.958 + 0.285i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(-0.3227657998 - 2.210394979i\) |
\(L(\frac12)\) |
\(\approx\) |
\(-0.3227657998 - 2.210394979i\) |
\(L(1)\) |
\(\approx\) |
\(1.197847672 - 0.8403810366i\) |
\(L(1)\) |
\(\approx\) |
\(1.197847672 - 0.8403810366i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (0.781 - 0.623i)T \) |
| 11 | \( 1 + (-0.988 + 0.149i)T \) |
| 13 | \( 1 + (-0.149 - 0.988i)T \) |
| 17 | \( 1 + (0.680 + 0.733i)T \) |
| 19 | \( 1 + (0.5 - 0.866i)T \) |
| 23 | \( 1 + (0.294 + 0.955i)T \) |
| 29 | \( 1 + (0.733 - 0.680i)T \) |
| 31 | \( 1 + T \) |
| 37 | \( 1 + (0.294 - 0.955i)T \) |
| 41 | \( 1 + (0.0747 - 0.997i)T \) |
| 43 | \( 1 + (-0.997 + 0.0747i)T \) |
| 47 | \( 1 + (0.781 - 0.623i)T \) |
| 53 | \( 1 + (0.294 + 0.955i)T \) |
| 59 | \( 1 + (0.900 + 0.433i)T \) |
| 61 | \( 1 + (-0.222 - 0.974i)T \) |
| 67 | \( 1 - iT \) |
| 71 | \( 1 + (-0.222 + 0.974i)T \) |
| 73 | \( 1 + (0.149 - 0.988i)T \) |
| 79 | \( 1 - T \) |
| 83 | \( 1 + (0.149 - 0.988i)T \) |
| 89 | \( 1 + (-0.365 - 0.930i)T \) |
| 97 | \( 1 + (-0.866 + 0.5i)T \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−20.26232237953867983945092751791, −19.07758963399609766076422470259, −18.42511798794816682381613592067, −17.76024450313003192069799300168, −16.57145692667015793549659939923, −16.468373688184470036318765715905, −15.64083915950557067758146444039, −14.77401943973898892578604708191, −14.16523230143490648751457052287, −13.56165603690948169468738894205, −12.76917616646846986771944308074, −11.99452290766305763886580519530, −11.48585625038674627041773007386, −10.39476518092974307903198233303, −9.63206224642101990440343481979, −8.4921752577620964472218907789, −8.026345864345136859529149953697, −7.07738059313415297519539486052, −6.49793146293080484470353917949, −5.52009558208036485214098037556, −4.89601968552119046359555301036, −4.1507595457421140322374850774, −3.05364198134192784595680792879, −2.54439094350491920860162876325, −1.19581477975109001753842522000,
0.29257614145230552110925677183, 1.14691115351053331361839635623, 2.316787727663059251350266052436, 2.96576825870959908011577270259, 3.75819552200104335603867425014, 4.78893883722240562013060187553, 5.421485595674644314516262348169, 6.04708068613585287738148315946, 7.17371001475664847053627693525, 7.864838555164069272658339366053, 8.89906074762861858969706509893, 9.95657922749890789872448853279, 10.333189266106454315195901537300, 11.13231249827143043516307466520, 11.96761894228775578138727637180, 12.61732813602148715697683811828, 13.33763569559840156144601268921, 13.828046743539617125340805152470, 14.839443183393686993775715675828, 15.4963734942802423941215738382, 15.84015971669288041892005134103, 17.13154402001615983991906469044, 17.81611291848878137626196423472, 18.59255671986775064102576243253, 19.36442277753347899295991606670