L(s) = 1 | + (0.987 − 0.156i)2-s + (−0.707 + 0.707i)3-s + (0.951 − 0.309i)4-s + (−0.587 + 0.809i)6-s + (−0.522 + 0.852i)7-s + (0.891 − 0.453i)8-s − i·9-s + (−0.233 + 0.972i)11-s + (−0.453 + 0.891i)12-s + (−0.0784 + 0.996i)13-s + (−0.382 + 0.923i)14-s + (0.809 − 0.587i)16-s + (0.0784 − 0.996i)17-s + (−0.156 − 0.987i)18-s + (0.0784 + 0.996i)19-s + ⋯ |
L(s) = 1 | + (0.987 − 0.156i)2-s + (−0.707 + 0.707i)3-s + (0.951 − 0.309i)4-s + (−0.587 + 0.809i)6-s + (−0.522 + 0.852i)7-s + (0.891 − 0.453i)8-s − i·9-s + (−0.233 + 0.972i)11-s + (−0.453 + 0.891i)12-s + (−0.0784 + 0.996i)13-s + (−0.382 + 0.923i)14-s + (0.809 − 0.587i)16-s + (0.0784 − 0.996i)17-s + (−0.156 − 0.987i)18-s + (0.0784 + 0.996i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.751 + 0.659i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.751 + 0.659i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7636363108 + 2.028481043i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7636363108 + 2.028481043i\) |
\(L(1)\) |
\(\approx\) |
\(1.330757651 + 0.5655290321i\) |
\(L(1)\) |
\(\approx\) |
\(1.330757651 + 0.5655290321i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 241 | \( 1 \) |
good | 2 | \( 1 + (0.987 - 0.156i)T \) |
| 3 | \( 1 + (-0.707 + 0.707i)T \) |
| 7 | \( 1 + (-0.522 + 0.852i)T \) |
| 11 | \( 1 + (-0.233 + 0.972i)T \) |
| 13 | \( 1 + (-0.0784 + 0.996i)T \) |
| 17 | \( 1 + (0.0784 - 0.996i)T \) |
| 19 | \( 1 + (0.0784 + 0.996i)T \) |
| 23 | \( 1 + (-0.522 + 0.852i)T \) |
| 29 | \( 1 + (0.453 + 0.891i)T \) |
| 31 | \( 1 + (-0.996 - 0.0784i)T \) |
| 37 | \( 1 + (0.972 + 0.233i)T \) |
| 41 | \( 1 + (0.891 - 0.453i)T \) |
| 43 | \( 1 + (0.852 - 0.522i)T \) |
| 47 | \( 1 + (0.891 - 0.453i)T \) |
| 53 | \( 1 + (0.453 + 0.891i)T \) |
| 59 | \( 1 + (0.156 + 0.987i)T \) |
| 61 | \( 1 + (-0.156 + 0.987i)T \) |
| 67 | \( 1 + (0.891 + 0.453i)T \) |
| 71 | \( 1 + (-0.760 - 0.649i)T \) |
| 73 | \( 1 + (-0.0784 - 0.996i)T \) |
| 79 | \( 1 + (-0.987 - 0.156i)T \) |
| 83 | \( 1 + T \) |
| 89 | \( 1 + (-0.852 + 0.522i)T \) |
| 97 | \( 1 + (0.309 - 0.951i)T \) |
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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
−17.29345819790061783575811434365, −16.79115507377024523830095002917, −16.03379228028792782540888425710, −15.72883679401479348752148682810, −14.60673856658976980840699221743, −14.08293007664743697772649033939, −13.30202582927710102138892646875, −12.89489437974281732490547276474, −12.54807410512837945425829513329, −11.533821437515276593161175446187, −10.87401058220034264884132621307, −10.65359607186362005813569699260, −9.70714787964659201799096248183, −8.31844612452568394881374775899, −7.90147859413730687466830212329, −7.19219383756391589323206654386, −6.43685150286995201451163675257, −5.981204906968024837675501049960, −5.40131684144501570608741707860, −4.469280160755381104774874727160, −3.861005033048461323287328225886, −2.9002404528637440287758388422, −2.358258641931670217350175352752, −1.15360820290318202091123929017, −0.43833189116335588621400627350,
1.20463439279186534206902557429, 2.1629550148876901397508653117, 2.838500517524419311688999600, 3.800186337824244442119507725837, 4.26375302560618629197627981019, 5.09529065782920924319509457788, 5.64872405684178950268251298452, 6.16246636971382016023326719270, 7.07206877077352356147632428206, 7.50873664628528449658459658291, 8.98514445602275104597271025140, 9.44795653186516559440525068753, 10.13058286269534491237449382615, 10.76023252239692594881531834942, 11.66018977981946086079229856661, 12.05383744332017475447342832495, 12.44454812983092399055609127151, 13.29060887307044408406975506861, 14.1792441843432030941532781029, 14.721334735185141214946461079070, 15.35194229783409111420406229, 16.037309582557270866589912594666, 16.30119283991752082649499195608, 17.03998759222493840459373873729, 18.08756853144512719167249150557