L(s) = 1 | + (0.766 − 0.642i)3-s + (0.939 + 0.342i)5-s + (0.5 − 0.866i)7-s + (0.173 − 0.984i)9-s + (−0.5 − 0.866i)11-s + (−0.766 − 0.642i)13-s + (0.939 − 0.342i)15-s + (0.173 + 0.984i)17-s + (−0.173 − 0.984i)21-s + (0.939 − 0.342i)23-s + (0.766 + 0.642i)25-s + (−0.5 − 0.866i)27-s + (−0.173 + 0.984i)29-s + (0.5 − 0.866i)31-s + (−0.939 − 0.342i)33-s + ⋯ |
L(s) = 1 | + (0.766 − 0.642i)3-s + (0.939 + 0.342i)5-s + (0.5 − 0.866i)7-s + (0.173 − 0.984i)9-s + (−0.5 − 0.866i)11-s + (−0.766 − 0.642i)13-s + (0.939 − 0.342i)15-s + (0.173 + 0.984i)17-s + (−0.173 − 0.984i)21-s + (0.939 − 0.342i)23-s + (0.766 + 0.642i)25-s + (−0.5 − 0.866i)27-s + (−0.173 + 0.984i)29-s + (0.5 − 0.866i)31-s + (−0.939 − 0.342i)33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.188 - 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.188 - 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.108265415 - 1.741996448i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.108265415 - 1.741996448i\) |
\(L(1)\) |
\(\approx\) |
\(1.523674609 - 0.5926414312i\) |
\(L(1)\) |
\(\approx\) |
\(1.523674609 - 0.5926414312i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 19 | \( 1 \) |
good | 3 | \( 1 + (0.766 - 0.642i)T \) |
| 5 | \( 1 + (0.939 + 0.342i)T \) |
| 7 | \( 1 + (0.5 - 0.866i)T \) |
| 11 | \( 1 + (-0.5 - 0.866i)T \) |
| 13 | \( 1 + (-0.766 - 0.642i)T \) |
| 17 | \( 1 + (0.173 + 0.984i)T \) |
| 23 | \( 1 + (0.939 - 0.342i)T \) |
| 29 | \( 1 + (-0.173 + 0.984i)T \) |
| 31 | \( 1 + (0.5 - 0.866i)T \) |
| 37 | \( 1 - T \) |
| 41 | \( 1 + (0.766 - 0.642i)T \) |
| 43 | \( 1 + (-0.939 - 0.342i)T \) |
| 47 | \( 1 + (-0.173 + 0.984i)T \) |
| 53 | \( 1 + (0.939 - 0.342i)T \) |
| 59 | \( 1 + (0.173 + 0.984i)T \) |
| 61 | \( 1 + (0.939 - 0.342i)T \) |
| 67 | \( 1 + (0.173 - 0.984i)T \) |
| 71 | \( 1 + (0.939 + 0.342i)T \) |
| 73 | \( 1 + (0.766 - 0.642i)T \) |
| 79 | \( 1 + (-0.766 + 0.642i)T \) |
| 83 | \( 1 + (-0.5 + 0.866i)T \) |
| 89 | \( 1 + (0.766 + 0.642i)T \) |
| 97 | \( 1 + (0.173 + 0.984i)T \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−28.050702265563405953753304797472, −26.950207413550705520040023350934, −26.00229065785450179969271602649, −24.98755854146173464891803267319, −24.63878147086585182799038056535, −22.93845640370368693683125935348, −21.71120384236465473669995796546, −21.170750241741520075179537638118, −20.381368031777013941001768287287, −19.12364147894802215531740687835, −18.038278131787555233638795503963, −16.99441559280151306214310292014, −15.79501018725553368787755033276, −14.84777667421419938400762546000, −13.99642085351810343176393037323, −12.90044360477577947369119267444, −11.64249385500772211338003031012, −10.09274321873780669644594459508, −9.423805803464026876053100471560, −8.47678063571413626698646304389, −7.117633934747334405250310958027, −5.30160910961958215889537137375, −4.700118545513268731546035192350, −2.754466858056584275864221377113, −1.89702399862915952820087819117,
0.97644975791048849645702335794, 2.33160263487458275828304517618, 3.50997031172828984595356144347, 5.30324374625113480792810330429, 6.624259743032591141027276594348, 7.67039629931103945059416793398, 8.69698169579601400481941378432, 10.045546017381901717558589394236, 10.92785470272299273040899470385, 12.639174669099420333655379988450, 13.4514960149497606701754771382, 14.28823908812190882616536292800, 15.09985640223512568448076854059, 16.87245340845466342631140238470, 17.642419514448354984315620568915, 18.65450968289720393135935399666, 19.60153466744318311059448834040, 20.720458877315422256589886997283, 21.40175741580903544402445343017, 22.72094797143233212391991552486, 24.00846006443220727677551641433, 24.54878199868584310517699050216, 25.73097724327480153015620130257, 26.392770303754277346737279018029, 27.29345364466320196428494035296