L(s) = 1 | + (0.844 − 0.535i)2-s + (−0.261 + 0.965i)3-s + (0.427 − 0.904i)4-s + (−0.825 − 0.564i)5-s + (0.295 + 0.955i)6-s + (−0.688 − 0.725i)7-s + (−0.123 − 0.992i)8-s + (−0.863 − 0.505i)9-s + (−0.999 − 0.0352i)10-s + (−0.579 − 0.815i)11-s + (0.760 + 0.648i)12-s + (0.0176 − 0.999i)13-s + (−0.969 − 0.244i)14-s + (0.760 − 0.648i)15-s + (−0.635 − 0.772i)16-s + (0.880 + 0.474i)17-s + ⋯ |
L(s) = 1 | + (0.844 − 0.535i)2-s + (−0.261 + 0.965i)3-s + (0.427 − 0.904i)4-s + (−0.825 − 0.564i)5-s + (0.295 + 0.955i)6-s + (−0.688 − 0.725i)7-s + (−0.123 − 0.992i)8-s + (−0.863 − 0.505i)9-s + (−0.999 − 0.0352i)10-s + (−0.579 − 0.815i)11-s + (0.760 + 0.648i)12-s + (0.0176 − 0.999i)13-s + (−0.969 − 0.244i)14-s + (0.760 − 0.648i)15-s + (−0.635 − 0.772i)16-s + (0.880 + 0.474i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 179 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.356 - 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 179 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.356 - 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6301571598 - 0.9144647577i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6301571598 - 0.9144647577i\) |
\(L(1)\) |
\(\approx\) |
\(1.021664299 - 0.4956146408i\) |
\(L(1)\) |
\(\approx\) |
\(1.021664299 - 0.4956146408i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 179 | \( 1 \) |
good | 2 | \( 1 + (0.844 - 0.535i)T \) |
| 3 | \( 1 + (-0.261 + 0.965i)T \) |
| 5 | \( 1 + (-0.825 - 0.564i)T \) |
| 7 | \( 1 + (-0.688 - 0.725i)T \) |
| 11 | \( 1 + (-0.579 - 0.815i)T \) |
| 13 | \( 1 + (0.0176 - 0.999i)T \) |
| 17 | \( 1 + (0.880 + 0.474i)T \) |
| 19 | \( 1 + (0.607 + 0.794i)T \) |
| 23 | \( 1 + (0.662 - 0.749i)T \) |
| 29 | \( 1 + (-0.783 + 0.621i)T \) |
| 31 | \( 1 + (-0.896 + 0.442i)T \) |
| 37 | \( 1 + (-0.783 - 0.621i)T \) |
| 41 | \( 1 + (0.911 + 0.411i)T \) |
| 43 | \( 1 + (0.362 - 0.932i)T \) |
| 47 | \( 1 + (0.0881 - 0.996i)T \) |
| 53 | \( 1 + (0.713 + 0.700i)T \) |
| 59 | \( 1 + (-0.737 - 0.675i)T \) |
| 61 | \( 1 + (-0.984 - 0.175i)T \) |
| 67 | \( 1 + (-0.123 + 0.992i)T \) |
| 71 | \( 1 + (-0.329 - 0.944i)T \) |
| 73 | \( 1 + (0.977 + 0.210i)T \) |
| 79 | \( 1 + (0.997 + 0.0705i)T \) |
| 83 | \( 1 + (0.550 - 0.835i)T \) |
| 89 | \( 1 + (0.844 + 0.535i)T \) |
| 97 | \( 1 + (0.990 - 0.140i)T \) |
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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
−27.73158965941525116005213572602, −26.049025481263645766463926553785, −25.79071238286097608440622058537, −24.52331816599414827089614721502, −23.75955508686439599717915849311, −22.87648262929664414397699273192, −22.42479193302977345332947058411, −21.109487372397832111506454253678, −19.77594241993248812125883617677, −18.8537975463515193894473745846, −17.98758011912849032434138512561, −16.68533364260704800992305152790, −15.735394159541298669673686131668, −14.85384324862704145294622858110, −13.741833416365919632153138555017, −12.72392169795805902340325017740, −11.93845885326930777300877387268, −11.19982635891943490399299627801, −9.20304516154919106858231153926, −7.64771314699293426128422815529, −7.18176169492355086851824278458, −6.090403889321776024642508229948, −4.941990337329764676345027753585, −3.33332227917359695142826941515, −2.3226601132837582966805883557,
0.69347620064815732064262727668, 3.30243635026352891253549503587, 3.67308657553394785707839874862, 5.06333533320852034038827462038, 5.85217663446623126054670313020, 7.55787611525281197416756883449, 9.04987095224155199403050168102, 10.40347845813840466732045202947, 10.83154994116126575269811432798, 12.18324576342136206295348573133, 12.94303811598797241979198343383, 14.23596809518933378171623680512, 15.27716388849484737712531086977, 16.18365517545702950729733894925, 16.72880668685882648050959508742, 18.636700787569710450620941495313, 19.75921325677041271242469606938, 20.44865638998100588679731345542, 21.17779620887638941695735342934, 22.38827045747025921882063498907, 23.06961310498964503764150944278, 23.72167086545080178566989866917, 24.95003499169824556924941731365, 26.336050199435979778692601895485, 27.290414482294749848602326714220