L(s) = 1 | + 5-s + 11-s + (−0.5 + 0.866i)13-s + (0.5 − 0.866i)17-s + (0.5 + 0.866i)19-s − 23-s + 25-s + (0.5 + 0.866i)29-s + (−0.5 − 0.866i)31-s + (0.5 + 0.866i)37-s + (0.5 − 0.866i)41-s + (−0.5 − 0.866i)43-s + (−0.5 + 0.866i)47-s + (0.5 − 0.866i)53-s + 55-s + ⋯ |
L(s) = 1 | + 5-s + 11-s + (−0.5 + 0.866i)13-s + (0.5 − 0.866i)17-s + (0.5 + 0.866i)19-s − 23-s + 25-s + (0.5 + 0.866i)29-s + (−0.5 − 0.866i)31-s + (0.5 + 0.866i)37-s + (0.5 − 0.866i)41-s + (−0.5 − 0.866i)43-s + (−0.5 + 0.866i)47-s + (0.5 − 0.866i)53-s + 55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.975 + 0.220i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.975 + 0.220i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.705467629 + 0.1903605810i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.705467629 + 0.1903605810i\) |
\(L(1)\) |
\(\approx\) |
\(1.314848881 + 0.06304660571i\) |
\(L(1)\) |
\(\approx\) |
\(1.314848881 + 0.06304660571i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 + (-0.5 + 0.866i)T \) |
| 17 | \( 1 + (0.5 - 0.866i)T \) |
| 19 | \( 1 + (0.5 + 0.866i)T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 + (0.5 + 0.866i)T \) |
| 31 | \( 1 + (-0.5 - 0.866i)T \) |
| 37 | \( 1 + (0.5 + 0.866i)T \) |
| 41 | \( 1 + (0.5 - 0.866i)T \) |
| 43 | \( 1 + (-0.5 - 0.866i)T \) |
| 47 | \( 1 + (-0.5 + 0.866i)T \) |
| 53 | \( 1 + (0.5 - 0.866i)T \) |
| 59 | \( 1 + (0.5 + 0.866i)T \) |
| 61 | \( 1 + (-0.5 + 0.866i)T \) |
| 67 | \( 1 + (-0.5 - 0.866i)T \) |
| 71 | \( 1 - T \) |
| 73 | \( 1 + (0.5 - 0.866i)T \) |
| 79 | \( 1 + (0.5 - 0.866i)T \) |
| 83 | \( 1 + (0.5 + 0.866i)T \) |
| 89 | \( 1 + (0.5 + 0.866i)T \) |
| 97 | \( 1 + (0.5 + 0.866i)T \) |
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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
−23.61491569480099947990763562987, −22.6039759424242281415884176122, −21.82621755328228602332037507818, −21.33330979350416265090086154499, −20.027292445778090555625397005040, −19.6464165572141068620868784662, −18.30358805912409651456443148789, −17.61217854243862545302067511016, −16.9873293248992349032376655996, −15.9880115927436116696572120105, −14.81064358732828833757386138812, −14.23403222986703921703563298215, −13.23584346279372429173479311490, −12.45315109809474981656236822516, −11.433986492391984697025521079374, −10.28772470276676758681994520133, −9.67815162821262465423254413197, −8.715105871307706953813662809027, −7.62428122733026323997456276428, −6.466870560126644635963282025643, −5.747261804206828838649970088609, −4.69126817525044601104573656545, −3.41803789353178875313193384381, −2.2848384245399547811006381459, −1.13688126697383077927639925154,
1.31181600399821342518218751340, 2.27558029285428899665603408804, 3.569309711924958997958899055886, 4.74046063146494021790598533328, 5.78049784321640526665880017580, 6.61847452657026656614585068453, 7.59508726398205579384468151453, 8.94784620383816079301220785399, 9.5869668635572266360690600500, 10.35665096764590260686244492264, 11.69230075799299086575366817881, 12.244731776199143996361801772230, 13.54357279283713222749474432997, 14.17810156760251536090378681576, 14.79883912559839633558009116664, 16.35979606748988792264600687702, 16.68992101826537722962690306098, 17.779434257759131054749476673574, 18.46529947999974804761263507599, 19.42863667476662507479694968465, 20.41838821514747700668499027652, 21.12505856635402429123891718073, 22.15045997613591008108860604119, 22.47167385734139687878518018034, 23.84221238741622727723025060222