L(s) = 1 | + (0.988 − 0.151i)2-s + (0.953 − 0.299i)4-s + (0.272 + 0.962i)5-s + (0.483 − 0.875i)7-s + (0.897 − 0.441i)8-s + (0.415 + 0.909i)10-s + (−0.948 − 0.318i)13-s + (0.345 − 0.938i)14-s + (0.820 − 0.572i)16-s + (0.610 + 0.791i)17-s + (−0.564 − 0.825i)19-s + (0.548 + 0.836i)20-s + (0.981 − 0.189i)23-s + (−0.851 + 0.524i)25-s + (−0.985 − 0.170i)26-s + ⋯ |
L(s) = 1 | + (0.988 − 0.151i)2-s + (0.953 − 0.299i)4-s + (0.272 + 0.962i)5-s + (0.483 − 0.875i)7-s + (0.897 − 0.441i)8-s + (0.415 + 0.909i)10-s + (−0.948 − 0.318i)13-s + (0.345 − 0.938i)14-s + (0.820 − 0.572i)16-s + (0.610 + 0.791i)17-s + (−0.564 − 0.825i)19-s + (0.548 + 0.836i)20-s + (0.981 − 0.189i)23-s + (−0.851 + 0.524i)25-s + (−0.985 − 0.170i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.946 - 0.321i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.946 - 0.321i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(3.295494148 - 0.5448818141i\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.295494148 - 0.5448818141i\) |
\(L(1)\) |
\(\approx\) |
\(2.144852278 - 0.1927262381i\) |
\(L(1)\) |
\(\approx\) |
\(2.144852278 - 0.1927262381i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (0.988 - 0.151i)T \) |
| 5 | \( 1 + (0.272 + 0.962i)T \) |
| 7 | \( 1 + (0.483 - 0.875i)T \) |
| 13 | \( 1 + (-0.948 - 0.318i)T \) |
| 17 | \( 1 + (0.610 + 0.791i)T \) |
| 19 | \( 1 + (-0.564 - 0.825i)T \) |
| 23 | \( 1 + (0.981 - 0.189i)T \) |
| 29 | \( 1 + (0.879 - 0.475i)T \) |
| 31 | \( 1 + (0.861 - 0.508i)T \) |
| 37 | \( 1 + (0.993 - 0.113i)T \) |
| 41 | \( 1 + (-0.935 + 0.353i)T \) |
| 43 | \( 1 + (-0.786 + 0.618i)T \) |
| 47 | \( 1 + (0.964 - 0.263i)T \) |
| 53 | \( 1 + (0.0855 - 0.996i)T \) |
| 59 | \( 1 + (0.161 + 0.986i)T \) |
| 61 | \( 1 + (-0.625 + 0.780i)T \) |
| 67 | \( 1 + (0.0475 + 0.998i)T \) |
| 71 | \( 1 + (0.941 + 0.336i)T \) |
| 73 | \( 1 + (-0.921 + 0.389i)T \) |
| 79 | \( 1 + (-0.532 + 0.846i)T \) |
| 83 | \( 1 + (-0.683 - 0.730i)T \) |
| 89 | \( 1 + (-0.959 - 0.281i)T \) |
| 97 | \( 1 + (0.272 - 0.962i)T \) |
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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
−21.51175282701043707813290524666, −20.91229475096652156133150071926, −20.22485377200379098574844307196, −19.31476854482500479317940032157, −18.42315447560237832963380974645, −17.185293472629273387956518924215, −16.83417955198680956688570246634, −15.86111116900021154655395243157, −15.18590041009558451789688559158, −14.32578438260219390892928599906, −13.73170406598046212563899341601, −12.61793411403123991168980790353, −12.211910614387527621785726717612, −11.60283341899760432651042206054, −10.422482704190713063593068593578, −9.43241702804085800718161141000, −8.501964947118068439921014921001, −7.73935656899243087197719960518, −6.67913087104959111549130914978, −5.71044660834011835630163087190, −4.99093482067853513510628859387, −4.53412842143452044666986864815, −3.16407280790188658704796209514, −2.260751426096055022714545910788, −1.33262465219549539610365091486,
1.13270549427785591856673702350, 2.37288000283945073191401401149, 3.02318787411973625027483900271, 4.11407472469929556284887811525, 4.82972931690927563069504469717, 5.869712176389909954212746675485, 6.76834463461385776159345913950, 7.32811715332351934353653458408, 8.26171818934841081891133276940, 9.96290572617696871479918813286, 10.30336524421014063228641036686, 11.18496671236514842183218116155, 11.82177928412309529229872078544, 12.98339547746274172352012035799, 13.49946289967823463456164488723, 14.48920359200717404034625247255, 14.82574423317217963050002507603, 15.56586785094860881176613032686, 16.9538648844158148039212294224, 17.20026220110888885612620231001, 18.38150064286772586968361801887, 19.42514733348374796436327731509, 19.77060080865107889329384597510, 20.90747732698354445924804417042, 21.45826942449262864120166853808