L(s) = 1 | + (−0.5 + 0.866i)3-s + (0.5 + 0.866i)5-s + (−0.5 − 0.866i)9-s + (−0.5 + 0.866i)11-s − 13-s − 15-s + (−0.5 + 0.866i)17-s + (−0.5 − 0.866i)19-s + (0.5 + 0.866i)23-s + (−0.5 + 0.866i)25-s + 27-s − 29-s + (0.5 − 0.866i)31-s + (−0.5 − 0.866i)33-s + (0.5 + 0.866i)37-s + ⋯ |
L(s) = 1 | + (−0.5 + 0.866i)3-s + (0.5 + 0.866i)5-s + (−0.5 − 0.866i)9-s + (−0.5 + 0.866i)11-s − 13-s − 15-s + (−0.5 + 0.866i)17-s + (−0.5 − 0.866i)19-s + (0.5 + 0.866i)23-s + (−0.5 + 0.866i)25-s + 27-s − 29-s + (0.5 − 0.866i)31-s + (−0.5 − 0.866i)33-s + (0.5 + 0.866i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 56 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.895 + 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 56 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.895 + 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2127780276 + 0.9082936223i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2127780276 + 0.9082936223i\) |
\(L(1)\) |
\(\approx\) |
\(0.6990897259 + 0.4650218252i\) |
\(L(1)\) |
\(\approx\) |
\(0.6990897259 + 0.4650218252i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + (-0.5 + 0.866i)T \) |
| 5 | \( 1 + (0.5 + 0.866i)T \) |
| 11 | \( 1 + (-0.5 + 0.866i)T \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 + (-0.5 + 0.866i)T \) |
| 19 | \( 1 + (-0.5 - 0.866i)T \) |
| 23 | \( 1 + (0.5 + 0.866i)T \) |
| 29 | \( 1 - T \) |
| 31 | \( 1 + (0.5 - 0.866i)T \) |
| 37 | \( 1 + (0.5 + 0.866i)T \) |
| 41 | \( 1 + T \) |
| 43 | \( 1 + T \) |
| 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 | \( 1 + T \) |
| 89 | \( 1 + (-0.5 - 0.866i)T \) |
| 97 | \( 1 + T \) |
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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
−32.13281654045398790134499577523, −31.23921960155871260564451031621, −29.6035910774765611977323694604, −29.168666954633157248086485137319, −28.04769879350391023623396315512, −26.67138757949227913581983410465, −24.958188249313275214275008582873, −24.51055603009769262657722873162, −23.31227544295145362654114420907, −22.024307826994827461401926752551, −20.74843583274657722691933261300, −19.4127170465784061773489912699, −18.25460211402449504551584536383, −17.086065714175798793766445583662, −16.234309791291067173955791344069, −14.2026002771023031059050468634, −13.06687760877605081775573291338, −12.16124496978078571724057788477, −10.69144534031462645991344060290, −8.983873631770688246855590808447, −7.63228666057880087569137432635, −6.06144687784873339162178666824, −4.91275094725652755567254160806, −2.31445603496412591753166182205, −0.526302844319604645236391096165,
2.561807585183171101102375643620, 4.38257876012493195080055360562, 5.81755931454230911076565056826, 7.24019005740766857875355064353, 9.36036697156193169166253895012, 10.326667724808013369870683634228, 11.391170592850916141809002293722, 13.01926126549215266048749945887, 14.76773452399755290162019886569, 15.376650802835246811951735895039, 17.12646220861935716819092767314, 17.78291035419521821393232715717, 19.38882017262599175102984463244, 20.86931957996837079506194929693, 21.908227826265944607319520075323, 22.66528232629425660293493084042, 23.94093297478523902248402235478, 25.72084177602388075666314901860, 26.388571914462508186307832720162, 27.60474158394939424828873065708, 28.71054655248816652230088619288, 29.71005535586695117929070605298, 31.02582026642803020475706988994, 32.31314427849658013068211840339, 33.44089760857108050533882887064