L(s) = 1 | + (−0.841 + 0.540i)2-s + (0.415 − 0.909i)4-s + (−0.959 + 0.281i)5-s + (0.654 + 0.755i)7-s + (0.142 + 0.989i)8-s + (0.654 − 0.755i)10-s + (0.841 + 0.540i)11-s + (−0.654 + 0.755i)13-s + (−0.959 − 0.281i)14-s + (−0.654 − 0.755i)16-s + (0.415 + 0.909i)17-s + (−0.415 + 0.909i)19-s + (−0.142 + 0.989i)20-s − 22-s + (0.841 − 0.540i)25-s + (0.142 − 0.989i)26-s + ⋯ |
L(s) = 1 | + (−0.841 + 0.540i)2-s + (0.415 − 0.909i)4-s + (−0.959 + 0.281i)5-s + (0.654 + 0.755i)7-s + (0.142 + 0.989i)8-s + (0.654 − 0.755i)10-s + (0.841 + 0.540i)11-s + (−0.654 + 0.755i)13-s + (−0.959 − 0.281i)14-s + (−0.654 − 0.755i)16-s + (0.415 + 0.909i)17-s + (−0.415 + 0.909i)19-s + (−0.142 + 0.989i)20-s − 22-s + (0.841 − 0.540i)25-s + (0.142 − 0.989i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0174 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0174 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4025361303 + 0.3955869120i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4025361303 + 0.3955869120i\) |
\(L(1)\) |
\(\approx\) |
\(0.5949408789 + 0.2848442032i\) |
\(L(1)\) |
\(\approx\) |
\(0.5949408789 + 0.2848442032i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 23 | \( 1 \) |
good | 2 | \( 1 + (-0.841 + 0.540i)T \) |
| 5 | \( 1 + (-0.959 + 0.281i)T \) |
| 7 | \( 1 + (0.654 + 0.755i)T \) |
| 11 | \( 1 + (0.841 + 0.540i)T \) |
| 13 | \( 1 + (-0.654 + 0.755i)T \) |
| 17 | \( 1 + (0.415 + 0.909i)T \) |
| 19 | \( 1 + (-0.415 + 0.909i)T \) |
| 29 | \( 1 + (-0.415 - 0.909i)T \) |
| 31 | \( 1 + (-0.142 - 0.989i)T \) |
| 37 | \( 1 + (0.959 + 0.281i)T \) |
| 41 | \( 1 + (0.959 - 0.281i)T \) |
| 43 | \( 1 + (0.142 - 0.989i)T \) |
| 47 | \( 1 - T \) |
| 53 | \( 1 + (-0.654 - 0.755i)T \) |
| 59 | \( 1 + (0.654 - 0.755i)T \) |
| 61 | \( 1 + (0.142 + 0.989i)T \) |
| 67 | \( 1 + (-0.841 + 0.540i)T \) |
| 71 | \( 1 + (-0.841 + 0.540i)T \) |
| 73 | \( 1 + (0.415 - 0.909i)T \) |
| 79 | \( 1 + (0.654 - 0.755i)T \) |
| 83 | \( 1 + (-0.959 - 0.281i)T \) |
| 89 | \( 1 + (-0.142 + 0.989i)T \) |
| 97 | \( 1 + (0.959 - 0.281i)T \) |
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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
−31.3430845113279519303180029222, −30.208241459046460146719113953898, −29.56092660430121307856580182689, −27.98430271987152079814774581171, −27.303043872864401340587312627477, −26.63130559142641400312276172636, −25.06655135307265171234600307696, −24.047334194174373917445420764512, −22.658254742334272563299960294032, −21.31199272048150534504053371367, −20.03047585727433679253436896171, −19.61144919105697733286695821933, −18.135885167084961781797801264763, −17.01711152352820173562617645204, −16.08484076873491499065353741260, −14.55911894465751850506128753576, −12.84483996906254130795022635093, −11.61839443657721991344326816885, −10.81919639883970437045592058789, −9.250872197448729643811651275840, −8.02116693167184603254743865921, −7.09213010458559384774309357540, −4.58448608098081294817842887779, −3.17563253665816250225279113004, −0.96232529288932026425437074967,
1.956854261784113560006561567285, 4.32241025153315076046534592527, 6.06698271625630670537347745582, 7.45485707256871330040134563943, 8.44037817829674051986894639637, 9.741135844622269325723308305130, 11.270206793107503950819598859753, 12.14607366258038728997767902013, 14.64395095540596760351072159392, 14.95041345179794517896176548209, 16.41461823817766312923896813622, 17.458426386629060991123161798753, 18.80957058643508463842610847725, 19.405072946220673782092382510867, 20.780721871934710206591127482906, 22.37006472174596003399796238367, 23.65342752155071504312291473991, 24.50891740711789645953770663627, 25.61891761786768845703201945154, 26.82930028618820665035514230325, 27.65048919783941068303471336221, 28.39297722297770265929463489229, 29.92172134467860378015308594987, 31.10041439932825405211259233870, 32.172162349357803818836520340809