L(s) = 1 | + (0.0806 − 0.996i)2-s + (−0.873 − 0.487i)3-s + (−0.986 − 0.160i)4-s + (0.813 + 0.581i)5-s + (−0.556 + 0.831i)6-s + (−0.0186 − 0.999i)7-s + (−0.239 + 0.970i)8-s + (0.524 + 0.851i)9-s + (0.645 − 0.763i)10-s + (−0.673 + 0.739i)11-s + (0.783 + 0.621i)12-s + (0.154 − 0.987i)13-s + (−0.998 − 0.0620i)14-s + (−0.426 − 0.904i)15-s + (0.948 + 0.317i)16-s + (0.664 − 0.747i)17-s + ⋯ |
L(s) = 1 | + (0.0806 − 0.996i)2-s + (−0.873 − 0.487i)3-s + (−0.986 − 0.160i)4-s + (0.813 + 0.581i)5-s + (−0.556 + 0.831i)6-s + (−0.0186 − 0.999i)7-s + (−0.239 + 0.970i)8-s + (0.524 + 0.851i)9-s + (0.645 − 0.763i)10-s + (−0.673 + 0.739i)11-s + (0.783 + 0.621i)12-s + (0.154 − 0.987i)13-s + (−0.998 − 0.0620i)14-s + (−0.426 − 0.904i)15-s + (0.948 + 0.317i)16-s + (0.664 − 0.747i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.855 - 0.518i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.855 - 0.518i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2537947196 - 0.9086369944i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2537947196 - 0.9086369944i\) |
\(L(1)\) |
\(\approx\) |
\(0.6319519487 - 0.5652358403i\) |
\(L(1)\) |
\(\approx\) |
\(0.6319519487 - 0.5652358403i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 23 | \( 1 \) |
good | 2 | \( 1 + (0.0806 - 0.996i)T \) |
| 3 | \( 1 + (-0.873 - 0.487i)T \) |
| 5 | \( 1 + (0.813 + 0.581i)T \) |
| 7 | \( 1 + (-0.0186 - 0.999i)T \) |
| 11 | \( 1 + (-0.673 + 0.739i)T \) |
| 13 | \( 1 + (0.154 - 0.987i)T \) |
| 17 | \( 1 + (0.664 - 0.747i)T \) |
| 19 | \( 1 + (-0.358 + 0.933i)T \) |
| 29 | \( 1 + (0.437 - 0.899i)T \) |
| 31 | \( 1 + (0.798 - 0.601i)T \) |
| 37 | \( 1 + (0.626 - 0.779i)T \) |
| 41 | \( 1 + (0.00620 - 0.999i)T \) |
| 43 | \( 1 + (-0.492 - 0.870i)T \) |
| 47 | \( 1 + (-0.0682 - 0.997i)T \) |
| 53 | \( 1 + (0.645 + 0.763i)T \) |
| 59 | \( 1 + (-0.806 - 0.591i)T \) |
| 61 | \( 1 + (-0.972 + 0.233i)T \) |
| 67 | \( 1 + (-0.966 + 0.257i)T \) |
| 71 | \( 1 + (0.912 + 0.409i)T \) |
| 73 | \( 1 + (0.437 + 0.899i)T \) |
| 79 | \( 1 + (0.717 + 0.696i)T \) |
| 83 | \( 1 + (-0.944 - 0.329i)T \) |
| 89 | \( 1 + (0.0310 - 0.999i)T \) |
| 97 | \( 1 + (0.275 - 0.961i)T \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−23.96023164110891967707510926236, −23.18533401586430980340708410296, −21.997094988560056674431984933159, −21.46695847202401028108909877088, −21.19037887713739885128450136809, −19.33296697194032152273806526122, −18.307134659582604622402800552286, −17.84510515782777320890299616271, −16.74773770119516750630259053946, −16.391661204902366146985197826617, −15.51786401926754432264966987586, −14.66814023360833529867359383447, −13.57609013392656655807518520545, −12.75671396620425794858186051343, −11.94615219573503434238982222874, −10.70096542061601461726039909753, −9.64787501203729550964439087635, −8.97231477998178333907459113729, −8.14948296039661030329698087570, −6.45910483438181481789274604446, −6.13366702427030231327033574615, −5.1118393980247376631307433409, −4.61322187906232101203605218511, −3.06606703977354811736449886962, −1.25048228823993936039870383944,
0.62710494527697155585117321137, 1.8129213558884010022512968687, 2.80106999458818659598285019083, 4.12006674790289206535394654179, 5.23295797644703571952061501677, 5.96260027642734201915541862394, 7.234909493088320105280336368778, 8.025013631734418521826924051438, 9.80122013502880286081638601318, 10.26730366870186415046077850909, 10.81594463227111495278715510039, 11.92480607552392431201851217547, 12.77692023651913417172849013999, 13.50018528694730113376909011168, 14.136266737723792347793142344538, 15.38168526046171347276662970191, 16.88348854713977672445548096722, 17.377585707769659812213062196363, 18.2584802277262407451658924110, 18.6654508998914766486514986685, 19.83935329556230458095850794027, 20.76553764850653086888757837327, 21.34378287205049627994558792561, 22.572971323980155635290987659120, 22.922230814670080081320403838773