L(s) = 1 | + (−0.994 + 0.104i)2-s + (−0.743 + 0.669i)3-s + (0.978 − 0.207i)4-s + (0.669 − 0.743i)6-s − i·7-s + (−0.951 + 0.309i)8-s + (0.104 − 0.994i)9-s + (−0.809 + 0.587i)11-s + (−0.587 + 0.809i)12-s + (−0.994 − 0.104i)13-s + (0.104 + 0.994i)14-s + (0.913 − 0.406i)16-s + (−0.207 + 0.978i)17-s + i·18-s + (0.669 + 0.743i)21-s + (0.743 − 0.669i)22-s + ⋯ |
L(s) = 1 | + (−0.994 + 0.104i)2-s + (−0.743 + 0.669i)3-s + (0.978 − 0.207i)4-s + (0.669 − 0.743i)6-s − i·7-s + (−0.951 + 0.309i)8-s + (0.104 − 0.994i)9-s + (−0.809 + 0.587i)11-s + (−0.587 + 0.809i)12-s + (−0.994 − 0.104i)13-s + (0.104 + 0.994i)14-s + (0.913 − 0.406i)16-s + (−0.207 + 0.978i)17-s + i·18-s + (0.669 + 0.743i)21-s + (0.743 − 0.669i)22-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.712 - 0.702i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.712 - 0.702i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3642637446 - 0.1493790799i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3642637446 - 0.1493790799i\) |
\(L(1)\) |
\(\approx\) |
\(0.4503122221 + 0.07561597076i\) |
\(L(1)\) |
\(\approx\) |
\(0.4503122221 + 0.07561597076i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 19 | \( 1 \) |
good | 2 | \( 1 + (-0.994 + 0.104i)T \) |
| 3 | \( 1 + (-0.743 + 0.669i)T \) |
| 7 | \( 1 - iT \) |
| 11 | \( 1 + (-0.809 + 0.587i)T \) |
| 13 | \( 1 + (-0.994 - 0.104i)T \) |
| 17 | \( 1 + (-0.207 + 0.978i)T \) |
| 23 | \( 1 + (-0.406 + 0.913i)T \) |
| 29 | \( 1 + (0.978 - 0.207i)T \) |
| 31 | \( 1 + (0.309 + 0.951i)T \) |
| 37 | \( 1 + (-0.587 + 0.809i)T \) |
| 41 | \( 1 + (0.913 - 0.406i)T \) |
| 43 | \( 1 + (-0.866 - 0.5i)T \) |
| 47 | \( 1 + (0.207 + 0.978i)T \) |
| 53 | \( 1 + (-0.207 - 0.978i)T \) |
| 59 | \( 1 + (-0.913 + 0.406i)T \) |
| 61 | \( 1 + (0.913 + 0.406i)T \) |
| 67 | \( 1 + (-0.743 - 0.669i)T \) |
| 71 | \( 1 + (0.669 + 0.743i)T \) |
| 73 | \( 1 + (0.994 - 0.104i)T \) |
| 79 | \( 1 + (-0.669 - 0.743i)T \) |
| 83 | \( 1 + (-0.951 + 0.309i)T \) |
| 89 | \( 1 + (-0.913 - 0.406i)T \) |
| 97 | \( 1 + (0.743 - 0.669i)T \) |
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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
−24.07304609831491722261584844046, −22.87528261944063971383093262960, −21.893218539862355726026287818331, −21.23623590117663693074271663126, −20.02620674513294219034986465675, −19.149342737843647429119971206194, −18.437210374316791648554066247723, −17.99767081053446628216652514083, −16.97797989244041605850598792256, −16.178585254670962982349972271743, −15.488149021636647106656223358310, −14.19849515298857029722653501918, −12.87567642881187475177618654296, −12.132946101845859399481302255352, −11.467874313250183356358531367522, −10.54268599582956452309773736259, −9.58344349738425632151518432573, −8.47763903971650227785434892364, −7.712762620871432980497588902202, −6.72955943930217398085445998500, −5.84186454957076318716639346855, −4.89771316142102350953439368853, −2.7515911407969182230095026476, −2.19517102598325799467790844342, −0.65544458427810985657354562829,
0.2510618622319499863686025449, 1.57409145976809325606900529584, 3.08594415260527440347084816563, 4.40219248952937194986184446606, 5.408499190509144966679865249416, 6.575853980927132423692745569060, 7.366348011449221118223746077, 8.358041527441592093522855181338, 9.69948302610517910035481499596, 10.20372883094569510925847025835, 10.822424195524983057527269247561, 11.8884056427976514711994975786, 12.73629720222728291013064627982, 14.233605362945589465309908533852, 15.294283733823957171421053813305, 15.85479024186115584701516981450, 16.880872308746775567529449093179, 17.45065737128251892571466727002, 17.97334699305112174009182713524, 19.338163742629273410829965646944, 19.999213969129587958567327027287, 20.91247875783001263185367629139, 21.59689986249433195092333864601, 22.763485894364738780189264284945, 23.66524088304814614611767195492