L(s) = 1 | + (0.104 − 0.994i)2-s + (0.978 + 0.207i)3-s + (−0.978 − 0.207i)4-s + (0.309 − 0.951i)6-s + (−0.309 + 0.951i)8-s + (0.913 + 0.406i)9-s + (0.913 − 0.406i)11-s + (−0.913 − 0.406i)12-s + (0.809 − 0.587i)13-s + (0.913 + 0.406i)16-s + (−0.669 − 0.743i)17-s + (0.5 − 0.866i)18-s + (−0.978 + 0.207i)19-s + (−0.309 − 0.951i)22-s + (0.104 − 0.994i)23-s + (−0.5 + 0.866i)24-s + ⋯ |
L(s) = 1 | + (0.104 − 0.994i)2-s + (0.978 + 0.207i)3-s + (−0.978 − 0.207i)4-s + (0.309 − 0.951i)6-s + (−0.309 + 0.951i)8-s + (0.913 + 0.406i)9-s + (0.913 − 0.406i)11-s + (−0.913 − 0.406i)12-s + (0.809 − 0.587i)13-s + (0.913 + 0.406i)16-s + (−0.669 − 0.743i)17-s + (0.5 − 0.866i)18-s + (−0.978 + 0.207i)19-s + (−0.309 − 0.951i)22-s + (0.104 − 0.994i)23-s + (−0.5 + 0.866i)24-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.309 - 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.309 - 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.245929270 - 0.9047035424i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.245929270 - 0.9047035424i\) |
\(L(1)\) |
\(\approx\) |
\(1.244373445 - 0.6173716568i\) |
\(L(1)\) |
\(\approx\) |
\(1.244373445 - 0.6173716568i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (0.104 - 0.994i)T \) |
| 3 | \( 1 + (0.978 + 0.207i)T \) |
| 11 | \( 1 + (0.913 - 0.406i)T \) |
| 13 | \( 1 + (0.809 - 0.587i)T \) |
| 17 | \( 1 + (-0.669 - 0.743i)T \) |
| 19 | \( 1 + (-0.978 + 0.207i)T \) |
| 23 | \( 1 + (0.104 - 0.994i)T \) |
| 29 | \( 1 + (0.309 + 0.951i)T \) |
| 31 | \( 1 + (0.669 + 0.743i)T \) |
| 37 | \( 1 + (-0.913 - 0.406i)T \) |
| 41 | \( 1 + (-0.809 + 0.587i)T \) |
| 43 | \( 1 - T \) |
| 47 | \( 1 + (-0.669 + 0.743i)T \) |
| 53 | \( 1 + (0.978 + 0.207i)T \) |
| 59 | \( 1 + (-0.104 - 0.994i)T \) |
| 61 | \( 1 + (-0.104 + 0.994i)T \) |
| 67 | \( 1 + (-0.669 - 0.743i)T \) |
| 71 | \( 1 + (0.309 + 0.951i)T \) |
| 73 | \( 1 + (-0.913 + 0.406i)T \) |
| 79 | \( 1 + (0.669 - 0.743i)T \) |
| 83 | \( 1 + (-0.309 + 0.951i)T \) |
| 89 | \( 1 + (-0.104 + 0.994i)T \) |
| 97 | \( 1 + (-0.309 - 0.951i)T \) |
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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
−27.36961525702346843170902601377, −26.28859610903040491477126558877, −25.71876018368626385063544484674, −24.85755241165538881462649596311, −24.0118684486917789501940591754, −23.11869442604195926618890196109, −21.865247816406610597264467556057, −21.00945617900975459565494713004, −19.618494299311783244974828709483, −18.91775135637475674346648879202, −17.73388697099076326363213229353, −16.8357769227722271941300842651, −15.450066980450256235240362218003, −14.98524738535652114359431587863, −13.773519877439955037177126430602, −13.22252658326908735397103567969, −11.88567387280759298845862942385, −10.04558119714133108116754087221, −8.94546434309461550533334033047, −8.317537521917977996016280734056, −6.99861556630840891605823108704, −6.2550960181692936103280631563, −4.43745176602604246516477279278, −3.6328491882607928321730224395, −1.73607757896762939303955112423,
1.41691159559981761751032487891, 2.79041811243786407248669885394, 3.76010102143533771476447429571, 4.84401508109833114283258721189, 6.59627806505841777797500189879, 8.419887469872464644490548090967, 8.86856129850319245503997527480, 10.14108359997346215795566103563, 11.01696708581199735466598142894, 12.33470134246102297632666800069, 13.3522770286168066680825494168, 14.15345172652931860793332729139, 15.07884994771223544472409601370, 16.39405799034525591568081502112, 17.84347574362127097337196170748, 18.79377457332074520750792070810, 19.68182224871867782280102698128, 20.409422369113264884556750368882, 21.271188849012286728226637301717, 22.16731406093359324259177638021, 23.15550286745664123608081249139, 24.477924819551278348267972124415, 25.40613969701044379148677922429, 26.560480968482734575985896801421, 27.3014631143640481526108347625