L(s) = 1 | + 2-s − 3-s + 4-s − i·5-s − 6-s + 7-s + 8-s + 9-s − i·10-s − i·11-s − 12-s + 13-s + 14-s + i·15-s + 16-s − 17-s + ⋯ |
L(s) = 1 | + 2-s − 3-s + 4-s − i·5-s − 6-s + 7-s + 8-s + 9-s − i·10-s − i·11-s − 12-s + 13-s + 14-s + i·15-s + 16-s − 17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0631 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0631 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.090951501 - 2.227356169i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.090951501 - 2.227356169i\) |
\(L(1)\) |
\(\approx\) |
\(1.655476337 - 0.5858910734i\) |
\(L(1)\) |
\(\approx\) |
\(1.655476337 - 0.5858910734i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 37 | \( 1 \) |
| 109 | \( 1 \) |
good | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 - iT \) |
| 17 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 + T \) |
| 31 | \( 1 - iT \) |
| 41 | \( 1 - T \) |
| 43 | \( 1 + T \) |
| 47 | \( 1 + T \) |
| 53 | \( 1 + iT \) |
| 59 | \( 1 + T \) |
| 61 | \( 1 - T \) |
| 67 | \( 1 + T \) |
| 71 | \( 1 + T \) |
| 73 | \( 1 - iT \) |
| 79 | \( 1 - T \) |
| 83 | \( 1 - iT \) |
| 89 | \( 1 - T \) |
| 97 | \( 1 - T \) |
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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
−18.39521685627135083846154917927, −17.79194477584855258982310407463, −17.573216707617021727809510462924, −16.38356262402991772655096743292, −15.69979743004202036728939475468, −15.37073731939690857104561459922, −14.45053904878409342970848149148, −13.95981141816212744898136782857, −13.219783268447352156622105177, −12.32693279527212748366985436867, −11.7451177949176194641361528461, −11.2639137459887776572703576426, −10.5079037367672080477457973451, −10.26440088825422401686682233109, −8.929224432752637193030932203412, −7.74879695610639612231990318036, −7.145438752821256737728885708865, −6.69704337325345000424908825455, −5.77272386393050678581117600837, −5.31343051689829365750993668233, −4.34282790602295059830130902380, −3.96331198823969572668670135799, −2.852578046200331793117522116521, −1.88340977029374999681945582185, −1.34854350304978191068700401155,
0.68181171960109391051471941791, 1.44696028133042260177035839714, 2.18943754287497879492942966119, 3.57560807627112439281188992435, 4.246572437495942049960568183229, 4.788171162490938795338804752423, 5.60655808577070738235035309983, 5.94951150554298837252283068374, 6.74544550247538582408092476727, 7.93182416734822488138260461328, 8.180920911989267487080921732042, 9.36201751806489152348444284150, 10.26944967193465721966092210522, 11.212085828683039393584760547686, 11.491391492023354261268363456610, 11.92240871332036789302336636406, 12.96835017551461108968004897061, 13.43722392738248818565177951193, 13.90002021406555345930244013980, 14.99524891519843040692277068292, 15.717969313110080464064153994640, 16.22212061577699381219917777224, 16.6786212707197648968587911009, 17.549237670622392320432605020884, 18.07533966261303419380283920422