L(s) = 1 | + (−0.104 − 0.994i)2-s + (−0.978 + 0.207i)4-s + (0.104 + 0.994i)5-s + (0.5 − 0.866i)7-s + (0.309 + 0.951i)8-s + (0.978 − 0.207i)10-s + (−0.669 − 0.743i)13-s + (−0.913 − 0.406i)14-s + (0.913 − 0.406i)16-s + (0.913 + 0.406i)17-s + (0.5 + 0.866i)19-s + (−0.309 − 0.951i)20-s + (−0.309 − 0.951i)23-s + (−0.978 + 0.207i)25-s + (−0.669 + 0.743i)26-s + ⋯ |
L(s) = 1 | + (−0.104 − 0.994i)2-s + (−0.978 + 0.207i)4-s + (0.104 + 0.994i)5-s + (0.5 − 0.866i)7-s + (0.309 + 0.951i)8-s + (0.978 − 0.207i)10-s + (−0.669 − 0.743i)13-s + (−0.913 − 0.406i)14-s + (0.913 − 0.406i)16-s + (0.913 + 0.406i)17-s + (0.5 + 0.866i)19-s + (−0.309 − 0.951i)20-s + (−0.309 − 0.951i)23-s + (−0.978 + 0.207i)25-s + (−0.669 + 0.743i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.243 - 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.243 - 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.125111890 - 0.8779599959i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.125111890 - 0.8779599959i\) |
\(L(1)\) |
\(\approx\) |
\(0.9187103479 - 0.4118278211i\) |
\(L(1)\) |
\(\approx\) |
\(0.9187103479 - 0.4118278211i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 11 | \( 1 \) |
| 61 | \( 1 \) |
good | 2 | \( 1 + (-0.104 - 0.994i)T \) |
| 5 | \( 1 + (0.104 + 0.994i)T \) |
| 7 | \( 1 + (0.5 - 0.866i)T \) |
| 13 | \( 1 + (-0.669 - 0.743i)T \) |
| 17 | \( 1 + (0.913 + 0.406i)T \) |
| 19 | \( 1 + (0.5 + 0.866i)T \) |
| 23 | \( 1 + (-0.309 - 0.951i)T \) |
| 29 | \( 1 + (-0.104 + 0.994i)T \) |
| 31 | \( 1 + (0.669 - 0.743i)T \) |
| 37 | \( 1 + T \) |
| 41 | \( 1 + (0.309 - 0.951i)T \) |
| 43 | \( 1 + (-0.669 - 0.743i)T \) |
| 47 | \( 1 + (-0.913 - 0.406i)T \) |
| 53 | \( 1 + (0.809 + 0.587i)T \) |
| 59 | \( 1 + (0.978 + 0.207i)T \) |
| 67 | \( 1 + (-0.978 - 0.207i)T \) |
| 71 | \( 1 + (0.5 + 0.866i)T \) |
| 73 | \( 1 + (-0.913 - 0.406i)T \) |
| 79 | \( 1 + (-0.913 + 0.406i)T \) |
| 83 | \( 1 + (0.669 + 0.743i)T \) |
| 89 | \( 1 + (0.809 + 0.587i)T \) |
| 97 | \( 1 + (0.669 - 0.743i)T \) |
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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
−19.86020378361351481518878805669, −19.32150641547981364728009928515, −18.43058016150901951531514360004, −17.70725796158209121163924688626, −17.21643240418515669620621309326, −16.25184065407995408013285016087, −15.97208553205344748127443777665, −14.97829121738582977736400413226, −14.43786201081849487471051670613, −13.53274787649764402921884567885, −12.9700084853690262992957476983, −11.876362736967026600547162142352, −11.61469152002866934593899781277, −9.913698022024373212548428484710, −9.51684151857015577927939560058, −8.817060301834492298001263863187, −8.01675992503731877103804488318, −7.471864105888336389127106993260, −6.349677567907965640942577577917, −5.60282316327652613161194018175, −4.88972677870649163258375464401, −4.44549433093436075847856459012, −3.11122213882926512987773154465, −1.852299970747840647195454371857, −0.87808618707043909710663646086,
0.69170974569425681012860006646, 1.74257910107983809603040408705, 2.64796944064844355948099543703, 3.459384174713043419837881411829, 4.11653885880546451940170903028, 5.14685944648745681126924897900, 5.96603654838445759386474635288, 7.21621777753746689119812416035, 7.78309895336041390146848364526, 8.51792992264702758591171545958, 9.86855975296456800123971285557, 10.17843704926086041141895532101, 10.73565630074093319800450914979, 11.60326684951753752150803445725, 12.26336847056436708513493172749, 13.12350227829081074189835533749, 13.94744734640301688332531842334, 14.529500141853732539211420917675, 14.990766808410670411384207064085, 16.44833775922237940235392516551, 17.06013358582232977410463562777, 17.8558202193670933883260544463, 18.38201190491105074317122588720, 19.067950547428214464049817553731, 19.85641599025145163632977675035