L(s) = 1 | + (−0.996 + 0.0868i)2-s + (0.0558 − 0.998i)3-s + (0.984 − 0.172i)4-s + (0.955 + 0.293i)5-s + (0.0310 + 0.999i)6-s + (0.735 + 0.678i)7-s + (−0.966 + 0.257i)8-s + (−0.993 − 0.111i)9-s + (−0.977 − 0.209i)10-s + (−0.287 + 0.957i)11-s + (−0.117 − 0.993i)12-s + (0.997 + 0.0744i)13-s + (−0.791 − 0.611i)14-s + (0.346 − 0.938i)15-s + (0.940 − 0.340i)16-s + (−0.709 + 0.704i)17-s + ⋯ |
L(s) = 1 | + (−0.996 + 0.0868i)2-s + (0.0558 − 0.998i)3-s + (0.984 − 0.172i)4-s + (0.955 + 0.293i)5-s + (0.0310 + 0.999i)6-s + (0.735 + 0.678i)7-s + (−0.966 + 0.257i)8-s + (−0.993 − 0.111i)9-s + (−0.977 − 0.209i)10-s + (−0.287 + 0.957i)11-s + (−0.117 − 0.993i)12-s + (0.997 + 0.0744i)13-s + (−0.791 − 0.611i)14-s + (0.346 − 0.938i)15-s + (0.940 − 0.340i)16-s + (−0.709 + 0.704i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.971 + 0.236i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.971 + 0.236i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.102822429 + 0.1322108577i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.102822429 + 0.1322108577i\) |
\(L(1)\) |
\(\approx\) |
\(0.8953581153 + 0.02808820189i\) |
\(L(1)\) |
\(\approx\) |
\(0.8953581153 + 0.02808820189i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 23 | \( 1 \) |
good | 2 | \( 1 + (-0.996 + 0.0868i)T \) |
| 3 | \( 1 + (0.0558 - 0.998i)T \) |
| 5 | \( 1 + (0.955 + 0.293i)T \) |
| 7 | \( 1 + (0.735 + 0.678i)T \) |
| 11 | \( 1 + (-0.287 + 0.957i)T \) |
| 13 | \( 1 + (0.997 + 0.0744i)T \) |
| 17 | \( 1 + (-0.709 + 0.704i)T \) |
| 19 | \( 1 + (-0.492 - 0.870i)T \) |
| 29 | \( 1 + (0.751 + 0.659i)T \) |
| 31 | \( 1 + (0.767 + 0.640i)T \) |
| 37 | \( 1 + (-0.358 - 0.933i)T \) |
| 41 | \( 1 + (0.969 - 0.245i)T \) |
| 43 | \( 1 + (-0.191 + 0.981i)T \) |
| 47 | \( 1 + (-0.917 + 0.398i)T \) |
| 53 | \( 1 + (-0.977 + 0.209i)T \) |
| 59 | \( 1 + (0.980 - 0.197i)T \) |
| 61 | \( 1 + (-0.999 - 0.0124i)T \) |
| 67 | \( 1 + (-0.535 - 0.844i)T \) |
| 71 | \( 1 + (-0.381 + 0.924i)T \) |
| 73 | \( 1 + (0.751 - 0.659i)T \) |
| 79 | \( 1 + (0.813 + 0.581i)T \) |
| 83 | \( 1 + (0.664 - 0.747i)T \) |
| 89 | \( 1 + (0.323 - 0.946i)T \) |
| 97 | \( 1 + (0.179 + 0.983i)T \) |
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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
−23.58157128437294899176624663491, −22.379480253174157528642099260804, −21.26090525557755137070707750725, −20.915192983370853805013371339, −20.394867810028036551877013814, −19.249099704176958970180548354048, −18.17609832241395140324493161353, −17.48470863265656027820066261317, −16.702744724064343293824803185717, −16.13093362796363592275293889171, −15.171286477706693960142040561881, −14.04531961212713567057663193098, −13.40217123208541982302650227655, −11.77418580654602078303004742197, −10.94872945785706765696503953651, −10.38127165368403390164112326786, −9.56100258589045198206384787056, −8.54482867860510193587669512536, −8.133432463964295868156167457090, −6.49488958483923992730540347935, −5.72528454719139561950552208627, −4.54080102325272847967846345971, −3.316918591713610920624225719234, −2.16684209048644301318427588925, −0.87368918132646075379818091530,
1.367932491728956811088555263792, 2.03845898489351690239630381888, 2.84251056284374966994422062471, 4.99857575800546383244628091903, 6.17865647136730097058740836680, 6.63442919087379723925905395372, 7.756763194945596572111539209651, 8.65194253065164305813881710940, 9.24594785706796970260191975997, 10.61069746792627432004125456632, 11.16481448806968065042703173291, 12.31198341764874092339524373316, 13.07894257411235863336975272040, 14.23777455569561925267225310533, 14.98148793234675064984304693577, 15.94207117894870982983741209120, 17.35972563355179638582231245241, 17.78487141553527796392561907898, 18.13705881507186993235693193241, 19.12060997323384120224513416431, 19.92080014630590990016772463845, 20.94593231232390960575006943962, 21.51209635490034552912692921161, 22.86087246598267640279476638938, 23.848772562769106388988387965303