L(s) = 1 | + (−0.358 − 0.933i)2-s + (0.940 + 0.340i)3-s + (−0.743 + 0.668i)4-s + (−0.691 − 0.722i)5-s + (−0.0186 − 0.999i)6-s + (0.901 + 0.432i)7-s + (0.890 + 0.454i)8-s + (0.767 + 0.640i)9-s + (−0.426 + 0.904i)10-s + (0.437 + 0.899i)11-s + (−0.926 + 0.375i)12-s + (−0.834 + 0.551i)13-s + (0.0806 − 0.996i)14-s + (−0.404 − 0.914i)15-s + (0.105 − 0.994i)16-s + (0.154 + 0.987i)17-s + ⋯ |
L(s) = 1 | + (−0.358 − 0.933i)2-s + (0.940 + 0.340i)3-s + (−0.743 + 0.668i)4-s + (−0.691 − 0.722i)5-s + (−0.0186 − 0.999i)6-s + (0.901 + 0.432i)7-s + (0.890 + 0.454i)8-s + (0.767 + 0.640i)9-s + (−0.426 + 0.904i)10-s + (0.437 + 0.899i)11-s + (−0.926 + 0.375i)12-s + (−0.834 + 0.551i)13-s + (0.0806 − 0.996i)14-s + (−0.404 − 0.914i)15-s + (0.105 − 0.994i)16-s + (0.154 + 0.987i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.951 + 0.307i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.951 + 0.307i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.291858481 + 0.2032163589i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.291858481 + 0.2032163589i\) |
\(L(1)\) |
\(\approx\) |
\(1.071109062 - 0.1294197464i\) |
\(L(1)\) |
\(\approx\) |
\(1.071109062 - 0.1294197464i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 23 | \( 1 \) |
good | 2 | \( 1 + (-0.358 - 0.933i)T \) |
| 3 | \( 1 + (0.940 + 0.340i)T \) |
| 5 | \( 1 + (-0.691 - 0.722i)T \) |
| 7 | \( 1 + (0.901 + 0.432i)T \) |
| 11 | \( 1 + (0.437 + 0.899i)T \) |
| 13 | \( 1 + (-0.834 + 0.551i)T \) |
| 17 | \( 1 + (0.154 + 0.987i)T \) |
| 19 | \( 1 + (-0.806 - 0.591i)T \) |
| 29 | \( 1 + (-0.117 + 0.993i)T \) |
| 31 | \( 1 + (-0.977 + 0.209i)T \) |
| 37 | \( 1 + (-0.860 + 0.508i)T \) |
| 41 | \( 1 + (0.988 - 0.148i)T \) |
| 43 | \( 1 + (0.980 - 0.197i)T \) |
| 47 | \( 1 + (-0.0682 + 0.997i)T \) |
| 53 | \( 1 + (-0.426 - 0.904i)T \) |
| 59 | \( 1 + (-0.873 - 0.487i)T \) |
| 61 | \( 1 + (0.813 - 0.581i)T \) |
| 67 | \( 1 + (0.999 - 0.0248i)T \) |
| 71 | \( 1 + (-0.759 + 0.650i)T \) |
| 73 | \( 1 + (-0.117 - 0.993i)T \) |
| 79 | \( 1 + (0.931 + 0.363i)T \) |
| 83 | \( 1 + (-0.191 - 0.981i)T \) |
| 89 | \( 1 + (0.735 - 0.678i)T \) |
| 97 | \( 1 + (0.912 - 0.409i)T \) |
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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
−23.64287789267977567505322085220, −22.916735101430767376276316218279, −21.90086391948008085461104948769, −20.76460877409026087059352970923, −19.745686438108006796522287209855, −19.15529906426172536800753975341, −18.42086435515041606453961523011, −17.62770336832894867605192402516, −16.62071392247402752033053521047, −15.605036425101414424060671736940, −14.78327243331490494042184259972, −14.32214672365622277129311943442, −13.643151775916184615480356915938, −12.366954379565053882453266877448, −11.15210378161649762900822620565, −10.24113740152657311394810403874, −9.159808403406301044658929436060, −8.21260793959710346535826813916, −7.6124529535822665325652274038, −7.01496205647390269298351849440, −5.81999038462633149392421978906, −4.44745424755569040878202610620, −3.61735393408253919118490947632, −2.256838225199698965268736358048, −0.738878170379410736414667417,
1.56042286614970837912894532761, 2.181000847318282187702700312737, 3.57196824197905622491170927220, 4.45901208789774809189504256487, 4.960173186276369714068359815231, 7.2413804481334758825382656325, 7.99747909051262361096527784592, 8.877626543320473733600211661847, 9.301416798108477413483766780186, 10.50165883351212188461716488569, 11.38281798471015981861207681234, 12.48230074025026538608033202282, 12.78389308750247874876720400976, 14.29417907449129629461899531033, 14.770975261582142929159232828931, 15.83042202681707886260688896225, 16.97831692274715065277968780017, 17.62661589890798021870252295827, 18.911519758633123453582290409803, 19.46128842092114740772330279232, 20.16019037465131543664664724682, 20.84518634861021888367759069763, 21.56748033525812675321057849195, 22.27661344660944594080822744300, 23.6611579054080721519272481923