L(s) = 1 | + (−0.169 − 0.985i)2-s + (−0.316 − 0.948i)3-s + (−0.942 + 0.334i)4-s + (−0.993 − 0.113i)5-s + (−0.881 + 0.472i)6-s + (0.672 + 0.739i)7-s + (0.489 + 0.872i)8-s + (−0.800 + 0.599i)9-s + (0.0567 + 0.998i)10-s + (−0.999 − 0.0378i)11-s + (0.614 + 0.788i)12-s + (−0.644 + 0.764i)13-s + (0.614 − 0.788i)14-s + (0.206 + 0.978i)15-s + (0.776 − 0.629i)16-s + (−0.965 + 0.261i)17-s + ⋯ |
L(s) = 1 | + (−0.169 − 0.985i)2-s + (−0.316 − 0.948i)3-s + (−0.942 + 0.334i)4-s + (−0.993 − 0.113i)5-s + (−0.881 + 0.472i)6-s + (0.672 + 0.739i)7-s + (0.489 + 0.872i)8-s + (−0.800 + 0.599i)9-s + (0.0567 + 0.998i)10-s + (−0.999 − 0.0378i)11-s + (0.614 + 0.788i)12-s + (−0.644 + 0.764i)13-s + (0.614 − 0.788i)14-s + (0.206 + 0.978i)15-s + (0.776 − 0.629i)16-s + (−0.965 + 0.261i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 167 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.750 + 0.661i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 167 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.750 + 0.661i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2475724293 + 0.09350634869i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2475724293 + 0.09350634869i\) |
\(L(1)\) |
\(\approx\) |
\(0.4692082786 - 0.2350623338i\) |
\(L(1)\) |
\(\approx\) |
\(0.4692082786 - 0.2350623338i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 167 | \( 1 \) |
good | 2 | \( 1 + (-0.169 - 0.985i)T \) |
| 3 | \( 1 + (-0.316 - 0.948i)T \) |
| 5 | \( 1 + (-0.993 - 0.113i)T \) |
| 7 | \( 1 + (0.672 + 0.739i)T \) |
| 11 | \( 1 + (-0.999 - 0.0378i)T \) |
| 13 | \( 1 + (-0.644 + 0.764i)T \) |
| 17 | \( 1 + (-0.965 + 0.261i)T \) |
| 19 | \( 1 + (0.954 + 0.298i)T \) |
| 23 | \( 1 + (-0.243 + 0.969i)T \) |
| 29 | \( 1 + (-0.243 - 0.969i)T \) |
| 31 | \( 1 + (-0.700 - 0.713i)T \) |
| 37 | \( 1 + (-0.800 - 0.599i)T \) |
| 41 | \( 1 + (-0.0189 + 0.999i)T \) |
| 43 | \( 1 + (0.898 + 0.438i)T \) |
| 47 | \( 1 + (-0.752 + 0.658i)T \) |
| 53 | \( 1 + (-0.982 + 0.188i)T \) |
| 59 | \( 1 + (-0.965 - 0.261i)T \) |
| 61 | \( 1 + (0.351 + 0.936i)T \) |
| 67 | \( 1 + (-0.993 + 0.113i)T \) |
| 71 | \( 1 + (-0.387 + 0.922i)T \) |
| 73 | \( 1 + (0.776 + 0.629i)T \) |
| 79 | \( 1 + (-0.455 - 0.890i)T \) |
| 83 | \( 1 + (-0.169 + 0.985i)T \) |
| 89 | \( 1 + (0.206 - 0.978i)T \) |
| 97 | \( 1 + (-0.700 + 0.713i)T \) |
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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
−27.17625824950508228512548002615, −26.73630501227460989152168670206, −25.957252805985385410932486565563, −24.3553412384737672795047044311, −23.7602170488014344106857275115, −22.75124553641941141441712187972, −22.139440599518665287329971454544, −20.57819820546519571268311512213, −19.86682333647869629713787420578, −18.284224581956443512846847427456, −17.53562568383085419478071952895, −16.40487677244288447615108988359, −15.69230924690040639159492495710, −14.93529821677197813913542131396, −13.955060621242614836553021599100, −12.459782227387731709232343501349, −10.94809524480344782569691806691, −10.32673484962703823957507175122, −8.88280745335079501579046050709, −7.86181914663772806867880650198, −6.9568347884727121274889977517, −5.19340573049148275026212177570, −4.66183795230752027786339663951, −3.36983513487834145857785907953, −0.241016074793398066809148796122,
1.67995509644276784912959876105, 2.78569614674192692820732697103, 4.45872518225275273941103844804, 5.552484949881252217913381252, 7.50017745163202197188230407731, 8.11422219270452485470132446350, 9.31623907787086850928729531581, 11.05067153505275286196691012521, 11.59589893337860911682258223173, 12.40615825846962333190686241910, 13.37018030065624474318192787550, 14.57362271468261780262435694898, 15.95411058528631339703864964917, 17.350279755216429202790053011497, 18.209028408554761276368511229386, 18.967933918587223829387840381357, 19.72923867566739746993401090932, 20.78759062353374804342771935882, 21.92918889395455152711454862928, 22.86871976209843727177517646280, 23.89855425017372788469645618388, 24.462703011649546412284272245508, 26.085468306751529325492328752884, 26.997958028453228343240741918805, 28.08969380710628534590017221343