L(s) = 1 | + (−0.836 − 0.548i)2-s + (−0.607 − 0.794i)3-s + (0.399 + 0.916i)4-s + (0.0241 + 0.999i)5-s + (0.0724 + 0.997i)6-s + (−0.644 + 0.764i)7-s + (0.168 − 0.985i)8-s + (−0.262 + 0.964i)9-s + (0.527 − 0.849i)10-s + (0.485 − 0.873i)12-s + (0.527 + 0.849i)13-s + (0.958 − 0.285i)14-s + (0.779 − 0.626i)15-s + (−0.681 + 0.732i)16-s + (−0.981 + 0.192i)17-s + (0.748 − 0.663i)18-s + ⋯ |
L(s) = 1 | + (−0.836 − 0.548i)2-s + (−0.607 − 0.794i)3-s + (0.399 + 0.916i)4-s + (0.0241 + 0.999i)5-s + (0.0724 + 0.997i)6-s + (−0.644 + 0.764i)7-s + (0.168 − 0.985i)8-s + (−0.262 + 0.964i)9-s + (0.527 − 0.849i)10-s + (0.485 − 0.873i)12-s + (0.527 + 0.849i)13-s + (0.958 − 0.285i)14-s + (0.779 − 0.626i)15-s + (−0.681 + 0.732i)16-s + (−0.981 + 0.192i)17-s + (0.748 − 0.663i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.401 - 0.916i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.401 - 0.916i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4067693198 - 0.2659565220i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4067693198 - 0.2659565220i\) |
\(L(1)\) |
\(\approx\) |
\(0.5006541767 - 0.05629350963i\) |
\(L(1)\) |
\(\approx\) |
\(0.5006541767 - 0.05629350963i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 \) |
| 131 | \( 1 \) |
good | 2 | \( 1 + (-0.836 - 0.548i)T \) |
| 3 | \( 1 + (-0.607 - 0.794i)T \) |
| 5 | \( 1 + (0.0241 + 0.999i)T \) |
| 7 | \( 1 + (-0.644 + 0.764i)T \) |
| 13 | \( 1 + (0.527 + 0.849i)T \) |
| 17 | \( 1 + (-0.981 + 0.192i)T \) |
| 19 | \( 1 + (-0.120 - 0.992i)T \) |
| 23 | \( 1 + (0.715 + 0.698i)T \) |
| 29 | \( 1 + (0.262 + 0.964i)T \) |
| 31 | \( 1 + (-0.443 - 0.896i)T \) |
| 37 | \( 1 + (0.215 - 0.976i)T \) |
| 41 | \( 1 + (0.681 + 0.732i)T \) |
| 43 | \( 1 + (0.943 - 0.331i)T \) |
| 47 | \( 1 + (-0.354 - 0.935i)T \) |
| 53 | \( 1 + (-0.809 + 0.587i)T \) |
| 59 | \( 1 + (-0.906 - 0.421i)T \) |
| 61 | \( 1 + (-0.309 - 0.951i)T \) |
| 67 | \( 1 + (-0.943 - 0.331i)T \) |
| 71 | \( 1 + (0.885 + 0.464i)T \) |
| 73 | \( 1 + (-0.309 + 0.951i)T \) |
| 79 | \( 1 + (-0.568 + 0.822i)T \) |
| 83 | \( 1 + (-0.995 - 0.0965i)T \) |
| 89 | \( 1 + (-0.809 - 0.587i)T \) |
| 97 | \( 1 + (0.926 + 0.377i)T \) |
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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
−20.66168583395202266213192093216, −19.9660402529745427511372119106, −19.23154686700360255546356441125, −18.09920004300872299961404065431, −17.43432596964319561292349730649, −16.89303601855764526209909773041, −16.15435709508487406369448587323, −15.82758545278601474436215521464, −14.97527891218335077012924745273, −13.95154657964685929981140795355, −13.01263175271997250923913924896, −12.19512293703885094502384623886, −11.07869916959671689650271597583, −10.53479775738156928188147599638, −9.79999802854669936609674329347, −9.09378594463056051844848539002, −8.39502890189798159672358011208, −7.44555336927196714455729912333, −6.32907134794528884146179686025, −5.884336437117750000102996996011, −4.828509598392445856577493701060, −4.19512257218191671170548899422, −2.96359139722732801650014171627, −1.33903063873042025821351964306, −0.56292314109295555931510639495,
0.24186080375587484061147510295, 1.59326992256925363547221048642, 2.38013338121106743212002668393, 3.06012314968644873145170207545, 4.23790053300357653649130955481, 5.68105772742321485747435255817, 6.61233435459508978137126271998, 6.90532254020688482432066411625, 7.84454720368656777952099086767, 8.9494319421864072522301913, 9.43188799667285249290505605631, 10.684840272145007388404696319292, 11.18016121827341159436105638370, 11.65816344563358629484662187243, 12.727038464789129943335430385, 13.180391796055330320035766492983, 14.15447035647681099581896361978, 15.42754274269834955406975431232, 15.94214798648923101390358091748, 16.933616638374843953578430924033, 17.62907844428962955642148127623, 18.37745729312052138540234337211, 18.68950349426591324484699841650, 19.50360350342730894622655999344, 19.910899259105206803193624329922