L(s) = 1 | + (0.735 + 0.677i)2-s + (−0.401 + 0.915i)3-s + (0.0825 + 0.996i)4-s + (−0.945 − 0.324i)5-s + (−0.915 + 0.401i)6-s + (−0.969 + 0.245i)7-s + (−0.614 + 0.789i)8-s + (−0.677 − 0.735i)9-s + (−0.475 − 0.879i)10-s + (−0.546 + 0.837i)11-s + (−0.945 − 0.324i)12-s + (0.324 − 0.945i)13-s + (−0.879 − 0.475i)14-s + (0.677 − 0.735i)15-s + (−0.986 + 0.164i)16-s + (0.945 + 0.324i)17-s + ⋯ |
L(s) = 1 | + (0.735 + 0.677i)2-s + (−0.401 + 0.915i)3-s + (0.0825 + 0.996i)4-s + (−0.945 − 0.324i)5-s + (−0.915 + 0.401i)6-s + (−0.969 + 0.245i)7-s + (−0.614 + 0.789i)8-s + (−0.677 − 0.735i)9-s + (−0.475 − 0.879i)10-s + (−0.546 + 0.837i)11-s + (−0.945 − 0.324i)12-s + (0.324 − 0.945i)13-s + (−0.879 − 0.475i)14-s + (0.677 − 0.735i)15-s + (−0.986 + 0.164i)16-s + (0.945 + 0.324i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 229 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.822 - 0.569i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 229 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.822 - 0.569i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2587151322 - 0.08085540843i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2587151322 - 0.08085540843i\) |
\(L(1)\) |
\(\approx\) |
\(0.6504551656 + 0.5287950232i\) |
\(L(1)\) |
\(\approx\) |
\(0.6504551656 + 0.5287950232i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 229 | \( 1 \) |
good | 2 | \( 1 + (0.735 + 0.677i)T \) |
| 3 | \( 1 + (-0.401 + 0.915i)T \) |
| 5 | \( 1 + (-0.945 - 0.324i)T \) |
| 7 | \( 1 + (-0.969 + 0.245i)T \) |
| 11 | \( 1 + (-0.546 + 0.837i)T \) |
| 13 | \( 1 + (0.324 - 0.945i)T \) |
| 17 | \( 1 + (0.945 + 0.324i)T \) |
| 19 | \( 1 + (0.945 - 0.324i)T \) |
| 23 | \( 1 + (-0.475 + 0.879i)T \) |
| 29 | \( 1 + (0.969 - 0.245i)T \) |
| 31 | \( 1 + (-0.837 - 0.546i)T \) |
| 37 | \( 1 + (-0.986 - 0.164i)T \) |
| 41 | \( 1 + (-0.735 - 0.677i)T \) |
| 43 | \( 1 + (-0.986 - 0.164i)T \) |
| 47 | \( 1 + (0.735 - 0.677i)T \) |
| 53 | \( 1 + (-0.401 + 0.915i)T \) |
| 59 | \( 1 + (0.164 + 0.986i)T \) |
| 61 | \( 1 + (-0.677 - 0.735i)T \) |
| 67 | \( 1 + (0.735 - 0.677i)T \) |
| 71 | \( 1 + (-0.546 - 0.837i)T \) |
| 73 | \( 1 + (0.614 - 0.789i)T \) |
| 79 | \( 1 + (0.969 + 0.245i)T \) |
| 83 | \( 1 + (-0.986 + 0.164i)T \) |
| 89 | \( 1 - iT \) |
| 97 | \( 1 + (-0.789 + 0.614i)T \) |
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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
−26.26241272201195594823378956483, −25.00423960999377548717670177901, −23.86270509620918648189984728412, −23.46658097100479541332409612133, −22.65604012298104063196744283941, −21.85453424898510251864109492671, −20.48156820280738700765465064125, −19.574191310048537866103650088354, −18.80805340446045574533337327684, −18.40386520363064224926843464641, −16.397054518073287951847484484685, −15.989867663659331734508073265532, −14.32520658168176576011225685070, −13.71928818965484796292849015927, −12.58318559650226396673100067893, −11.92487073810885118146516367431, −11.06104646485385255924143342165, −10.06451178020846962983604922989, −8.44967630212849386119572825493, −7.1016635895467828255377099910, −6.32747041655134664130891941369, −5.1444507261276316772702963643, −3.62145621736649849815632437285, −2.82492738946957058632792988382, −1.112759904766906114522779176125,
0.08337063808068159701488847386, 3.133378844077320813719264667521, 3.75575906026728135425500100698, 5.04237832740185225313158007636, 5.742943882656059535863988146941, 7.14615812150090517262017389634, 8.166145179709049194733443218263, 9.390759084837598603598024635664, 10.54018239176151935265271369719, 11.96291469701072721267348232021, 12.412117453103459529304271003807, 13.643774522176376812227564948139, 15.188521699296088700367775022898, 15.49658443318177243026750397265, 16.23147082083761049622038323615, 17.15099731993252313016337342947, 18.28921348328722757255545964253, 19.91710610251678948876772142791, 20.55019104001432144482621074535, 21.66464283874422037236206480505, 22.612587972496302034273669975731, 23.11609106730454974635140465020, 23.85041786705217706896043548439, 25.28904564895792982086671808955, 25.90903944330878721654507749541