L(s) = 1 | + (−0.822 + 0.568i)2-s + (0.663 − 0.748i)3-s + (0.354 − 0.935i)4-s + (0.992 − 0.120i)5-s + (−0.120 + 0.992i)6-s + (−0.568 − 0.822i)7-s + (0.239 + 0.970i)8-s + (−0.120 − 0.992i)9-s + (−0.748 + 0.663i)10-s + (−0.885 + 0.464i)11-s + (−0.464 − 0.885i)12-s + (−0.354 − 0.935i)13-s + (0.935 + 0.354i)14-s + (0.568 − 0.822i)15-s + (−0.748 − 0.663i)16-s + (0.970 + 0.239i)17-s + ⋯ |
L(s) = 1 | + (−0.822 + 0.568i)2-s + (0.663 − 0.748i)3-s + (0.354 − 0.935i)4-s + (0.992 − 0.120i)5-s + (−0.120 + 0.992i)6-s + (−0.568 − 0.822i)7-s + (0.239 + 0.970i)8-s + (−0.120 − 0.992i)9-s + (−0.748 + 0.663i)10-s + (−0.885 + 0.464i)11-s + (−0.464 − 0.885i)12-s + (−0.354 − 0.935i)13-s + (0.935 + 0.354i)14-s + (0.568 − 0.822i)15-s + (−0.748 − 0.663i)16-s + (0.970 + 0.239i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 53 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.363 - 0.931i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 53 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.363 - 0.931i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.082198010 - 0.7394831192i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.082198010 - 0.7394831192i\) |
\(L(1)\) |
\(\approx\) |
\(0.9576680180 - 0.2541884710i\) |
\(L(1)\) |
\(\approx\) |
\(0.9576680180 - 0.2541884710i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 53 | \( 1 \) |
good | 2 | \( 1 + (-0.822 + 0.568i)T \) |
| 3 | \( 1 + (0.663 - 0.748i)T \) |
| 5 | \( 1 + (0.992 - 0.120i)T \) |
| 7 | \( 1 + (-0.568 - 0.822i)T \) |
| 11 | \( 1 + (-0.885 + 0.464i)T \) |
| 13 | \( 1 + (-0.354 - 0.935i)T \) |
| 17 | \( 1 + (0.970 + 0.239i)T \) |
| 19 | \( 1 + (0.935 - 0.354i)T \) |
| 23 | \( 1 - iT \) |
| 29 | \( 1 + (-0.885 - 0.464i)T \) |
| 31 | \( 1 + (-0.464 + 0.885i)T \) |
| 37 | \( 1 + (0.748 + 0.663i)T \) |
| 41 | \( 1 + (0.464 + 0.885i)T \) |
| 43 | \( 1 + (0.748 - 0.663i)T \) |
| 47 | \( 1 + (0.120 - 0.992i)T \) |
| 59 | \( 1 + (-0.120 + 0.992i)T \) |
| 61 | \( 1 + (0.239 + 0.970i)T \) |
| 67 | \( 1 + (0.935 + 0.354i)T \) |
| 71 | \( 1 + (0.663 + 0.748i)T \) |
| 73 | \( 1 + (0.239 - 0.970i)T \) |
| 79 | \( 1 + (0.822 + 0.568i)T \) |
| 83 | \( 1 + iT \) |
| 89 | \( 1 + (-0.970 - 0.239i)T \) |
| 97 | \( 1 + (0.120 + 0.992i)T \) |
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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
−33.46102555905908886112941191868, −31.91320678011426721755146081805, −31.15055070219721871159592794955, −29.53709560808093034817189446974, −28.75294736545697723140247755137, −27.63174107615770997295522004479, −26.28408870490182264834746400188, −25.77603554208259811191598071242, −24.699682229378329113174739592365, −22.22186672369239418752834179446, −21.448685701844394757606168225980, −20.6687205045737205597910990928, −19.166751516580481164265645915355, −18.37034521126047934880313710905, −16.73513982416107714004246693990, −15.83205573672481818578414747702, −14.15339486553868711798181829634, −12.82687378084267714062093616394, −11.13151666851765269035800764882, −9.6700892595449286661752509236, −9.31163075774159875969634798544, −7.678577346969500808366659430876, −5.56898932954641510607008964263, −3.30160536662122313040396340967, −2.15927970719026244139059752209,
0.92264801592774596852173774411, 2.649193389629308222487766446706, 5.55378303336925183591242876849, 6.968575709745799876658426110359, 7.97321113119882199820814360341, 9.52772643783075476659383619720, 10.36747400121802179859078265537, 12.736476138845187430396215034730, 13.805960650759414338441197164387, 14.97354334818028254079554591892, 16.56896813716249751866595540728, 17.72717800253545028842537958062, 18.5505738190513770281152315125, 19.954518924786091126607054186475, 20.68841371105137054734947279083, 22.86745409143687848486317391885, 24.04984876803371802184300581784, 25.13306812246985615614925469278, 25.88169467973248026302705644192, 26.71442001887095179118663088713, 28.49539433306440315501181826948, 29.34753260148508974913617987935, 30.28489048143074373681305437764, 32.146688507077753090221796901, 32.80910128692733571925795229390