L(s) = 1 | + (0.215 − 0.976i)2-s + (−0.262 + 0.964i)3-s + (−0.906 − 0.421i)4-s + (−0.998 + 0.0483i)5-s + (0.885 + 0.464i)6-s + (0.715 + 0.698i)7-s + (−0.607 + 0.794i)8-s + (−0.861 − 0.506i)9-s + (−0.168 + 0.985i)10-s + (0.644 − 0.764i)12-s + (−0.989 − 0.144i)13-s + (0.836 − 0.548i)14-s + (0.215 − 0.976i)15-s + (0.644 + 0.764i)16-s + (−0.970 − 0.239i)17-s + (−0.681 + 0.732i)18-s + ⋯ |
L(s) = 1 | + (0.215 − 0.976i)2-s + (−0.262 + 0.964i)3-s + (−0.906 − 0.421i)4-s + (−0.998 + 0.0483i)5-s + (0.885 + 0.464i)6-s + (0.715 + 0.698i)7-s + (−0.607 + 0.794i)8-s + (−0.861 − 0.506i)9-s + (−0.168 + 0.985i)10-s + (0.644 − 0.764i)12-s + (−0.989 − 0.144i)13-s + (0.836 − 0.548i)14-s + (0.215 − 0.976i)15-s + (0.644 + 0.764i)16-s + (−0.970 − 0.239i)17-s + (−0.681 + 0.732i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.986 - 0.164i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.986 - 0.164i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8514767959 - 0.07070572273i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8514767959 - 0.07070572273i\) |
\(L(1)\) |
\(\approx\) |
\(0.7601131412 - 0.1239439312i\) |
\(L(1)\) |
\(\approx\) |
\(0.7601131412 - 0.1239439312i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 \) |
| 131 | \( 1 \) |
good | 2 | \( 1 + (0.215 - 0.976i)T \) |
| 3 | \( 1 + (-0.262 + 0.964i)T \) |
| 5 | \( 1 + (-0.998 + 0.0483i)T \) |
| 7 | \( 1 + (0.715 + 0.698i)T \) |
| 13 | \( 1 + (-0.989 - 0.144i)T \) |
| 17 | \( 1 + (-0.970 - 0.239i)T \) |
| 19 | \( 1 + (0.644 - 0.764i)T \) |
| 23 | \( 1 + (-0.607 - 0.794i)T \) |
| 29 | \( 1 + (0.399 - 0.916i)T \) |
| 31 | \( 1 + (-0.607 + 0.794i)T \) |
| 37 | \( 1 + (0.981 - 0.192i)T \) |
| 41 | \( 1 + (-0.970 + 0.239i)T \) |
| 43 | \( 1 + (0.836 + 0.548i)T \) |
| 47 | \( 1 + (0.215 - 0.976i)T \) |
| 53 | \( 1 + (-0.809 + 0.587i)T \) |
| 59 | \( 1 + (-0.527 + 0.849i)T \) |
| 61 | \( 1 + T \) |
| 67 | \( 1 + (0.836 - 0.548i)T \) |
| 71 | \( 1 + (-0.607 + 0.794i)T \) |
| 73 | \( 1 + (0.309 - 0.951i)T \) |
| 79 | \( 1 + (-0.998 + 0.0483i)T \) |
| 83 | \( 1 + (0.981 + 0.192i)T \) |
| 89 | \( 1 + (-0.809 + 0.587i)T \) |
| 97 | \( 1 + (0.715 + 0.698i)T \) |
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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
−20.51704942475936119613082153933, −19.913738426178554321150762923663, −19.086046199488936698226817362988, −18.33624400426977397649877084636, −17.59726960249843201065302943452, −17.03118183288836050022333254609, −16.29354659906445321106321522893, −15.47538251448851941554943308964, −14.48457069741682773045875986171, −14.15431838034349470332454685628, −13.148018591848504057056183565647, −12.4788905945308775623737342172, −11.70746788130124070017149136867, −11.0343139011917498168056415426, −9.79220409329756728073503799984, −8.62951940237339694369378169721, −7.95085943758178858740194244826, −7.3974450390746684688198093651, −6.904382006084422244137245819865, −5.82232011590009364195623001143, −4.92872517148802178724333569186, −4.21790576017831758187810886706, −3.23285405562670336982942166607, −1.80706134553775944388046965533, −0.556682106157914381106255951635,
0.63093756928433105548189929928, 2.30540762219904073663291925252, 2.922572053023809101822372387577, 4.00605732557645305585559194350, 4.71976382328208948208082943867, 5.11469041301798867221575424149, 6.289533170709921568461642763144, 7.66376528915395043840353419146, 8.59734784605416233983366417025, 9.15214451235911733428741115045, 10.06945762446510309553240404066, 10.89363542455036390042108916656, 11.5367776244980711241088633777, 11.95398859809849398684216658462, 12.761164788262932991681412802662, 14.02078151208151419458183460803, 14.70420984849810264019567617424, 15.31343315756327673690799095070, 15.94551995058387617057300525820, 17.05931553803065677903081322302, 17.87113279814129775343757705680, 18.44810752200783260798827212156, 19.567236987488699906470852510190, 20.03122006766563647231486118222, 20.62872379850015131227005707529