L(s) = 1 | + (0.374 + 0.927i)2-s + (−0.559 − 0.829i)3-s + (−0.719 + 0.694i)4-s + (0.559 − 0.829i)6-s + (−0.5 + 0.866i)7-s + (−0.913 − 0.406i)8-s + (−0.374 + 0.927i)9-s + (−0.978 + 0.207i)11-s + (0.978 + 0.207i)12-s + (0.615 + 0.788i)13-s + (−0.990 − 0.139i)14-s + (0.0348 − 0.999i)16-s + (−0.241 + 0.970i)17-s − 18-s + (0.997 − 0.0697i)21-s + (−0.559 − 0.829i)22-s + ⋯ |
L(s) = 1 | + (0.374 + 0.927i)2-s + (−0.559 − 0.829i)3-s + (−0.719 + 0.694i)4-s + (0.559 − 0.829i)6-s + (−0.5 + 0.866i)7-s + (−0.913 − 0.406i)8-s + (−0.374 + 0.927i)9-s + (−0.978 + 0.207i)11-s + (0.978 + 0.207i)12-s + (0.615 + 0.788i)13-s + (−0.990 − 0.139i)14-s + (0.0348 − 0.999i)16-s + (−0.241 + 0.970i)17-s − 18-s + (0.997 − 0.0697i)21-s + (−0.559 − 0.829i)22-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.343 - 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.343 - 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.01814212901 + 0.01267995544i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.01814212901 + 0.01267995544i\) |
\(L(1)\) |
\(\approx\) |
\(0.6207423989 + 0.3602413192i\) |
\(L(1)\) |
\(\approx\) |
\(0.6207423989 + 0.3602413192i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 19 | \( 1 \) |
good | 2 | \( 1 + (0.374 + 0.927i)T \) |
| 3 | \( 1 + (-0.559 - 0.829i)T \) |
| 7 | \( 1 + (-0.5 + 0.866i)T \) |
| 11 | \( 1 + (-0.978 + 0.207i)T \) |
| 13 | \( 1 + (0.615 + 0.788i)T \) |
| 17 | \( 1 + (-0.241 + 0.970i)T \) |
| 23 | \( 1 + (-0.882 - 0.469i)T \) |
| 29 | \( 1 + (0.241 + 0.970i)T \) |
| 31 | \( 1 + (0.104 + 0.994i)T \) |
| 37 | \( 1 + (-0.309 - 0.951i)T \) |
| 41 | \( 1 + (-0.0348 + 0.999i)T \) |
| 43 | \( 1 + (0.173 + 0.984i)T \) |
| 47 | \( 1 + (-0.241 - 0.970i)T \) |
| 53 | \( 1 + (0.719 - 0.694i)T \) |
| 59 | \( 1 + (-0.848 - 0.529i)T \) |
| 61 | \( 1 + (-0.882 - 0.469i)T \) |
| 67 | \( 1 + (0.997 + 0.0697i)T \) |
| 71 | \( 1 + (-0.438 + 0.898i)T \) |
| 73 | \( 1 + (-0.615 + 0.788i)T \) |
| 79 | \( 1 + (-0.559 - 0.829i)T \) |
| 83 | \( 1 + (-0.104 - 0.994i)T \) |
| 89 | \( 1 + (-0.0348 - 0.999i)T \) |
| 97 | \( 1 + (0.997 - 0.0697i)T \) |
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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
−23.421300149619097065845872914913, −22.7009139950914502757409788189, −22.23128396527066613928146388760, −21.01907965187759367090562227866, −20.61797746356115032035036342758, −19.866123455064062257909633531469, −18.669558919561990141797469190372, −17.87869022343686827311126606677, −16.98380704803099805175309117984, −15.79683199068351461725974588818, −15.35102501618186276696065764834, −13.87458443164249250898895028460, −13.40649168659846101896351680934, −12.28230330566281795501039961314, −11.363233367024713970841756855268, −10.52460202140469172316228178346, −10.05519142356134264402558059276, −9.10531766687334354463251269900, −7.79335820058693194619431028281, −6.234486378059743641625082204465, −5.44415112484897424169274815633, −4.42865395903240294767024723660, −3.58654728495829934097395445393, −2.663811914536806587246076499687, −0.82118536579779465926376423127,
0.007275262509087898840409778771, 1.8666859342484551976818145950, 3.12616341792656030255784457945, 4.56654222228423138394824542752, 5.5834238072861829318693616820, 6.27229318115287951916150647129, 7.03150619688552922128106096215, 8.18674051872832456828413435930, 8.783211019520561368542773400370, 10.18811536132808273762876126414, 11.437731401079580832932268584024, 12.50825805962176445356610285422, 12.86215016055640929417550150068, 13.85442742305288456650038897522, 14.78975060664873229343376389211, 15.99336244671539170426968042007, 16.27439345684482425150739579119, 17.510151010349304340037410829502, 18.24524140492499926507170484592, 18.74467334771761148780322322637, 19.85613715389593094935262881925, 21.47230398639499913686622957668, 21.78302203354982566247589200003, 23.02511578299332777831568952070, 23.37389517861847857423231152774