L(s) = 1 | + (0.173 − 0.984i)2-s + (−0.866 − 0.5i)3-s + (−0.939 − 0.342i)4-s + (0.819 + 0.573i)5-s + (−0.642 + 0.766i)6-s + (0.258 + 0.965i)7-s + (−0.5 + 0.866i)8-s + (0.5 + 0.866i)9-s + (0.707 − 0.707i)10-s + (−0.573 + 0.819i)11-s + (0.642 + 0.766i)12-s + (−0.0871 − 0.996i)13-s + (0.996 − 0.0871i)14-s + (−0.422 − 0.906i)15-s + (0.766 + 0.642i)16-s + (0.965 + 0.258i)17-s + ⋯ |
L(s) = 1 | + (0.173 − 0.984i)2-s + (−0.866 − 0.5i)3-s + (−0.939 − 0.342i)4-s + (0.819 + 0.573i)5-s + (−0.642 + 0.766i)6-s + (0.258 + 0.965i)7-s + (−0.5 + 0.866i)8-s + (0.5 + 0.866i)9-s + (0.707 − 0.707i)10-s + (−0.573 + 0.819i)11-s + (0.642 + 0.766i)12-s + (−0.0871 − 0.996i)13-s + (0.996 − 0.0871i)14-s + (−0.422 − 0.906i)15-s + (0.766 + 0.642i)16-s + (0.965 + 0.258i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 73 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.997 - 0.0652i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 73 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.997 - 0.0652i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.228272370 - 0.04009230928i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.228272370 - 0.04009230928i\) |
\(L(1)\) |
\(\approx\) |
\(0.9092458760 - 0.2608289407i\) |
\(L(1)\) |
\(\approx\) |
\(0.9092458760 - 0.2608289407i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 73 | \( 1 \) |
good | 2 | \( 1 + (0.173 - 0.984i)T \) |
| 3 | \( 1 + (-0.866 - 0.5i)T \) |
| 5 | \( 1 + (0.819 + 0.573i)T \) |
| 7 | \( 1 + (0.258 + 0.965i)T \) |
| 11 | \( 1 + (-0.573 + 0.819i)T \) |
| 13 | \( 1 + (-0.0871 - 0.996i)T \) |
| 17 | \( 1 + (0.965 + 0.258i)T \) |
| 19 | \( 1 + (0.984 + 0.173i)T \) |
| 23 | \( 1 + (-0.984 + 0.173i)T \) |
| 29 | \( 1 + (-0.819 + 0.573i)T \) |
| 31 | \( 1 + (0.906 + 0.422i)T \) |
| 37 | \( 1 + (0.173 + 0.984i)T \) |
| 41 | \( 1 + (-0.766 + 0.642i)T \) |
| 43 | \( 1 + (0.965 - 0.258i)T \) |
| 47 | \( 1 + (0.996 + 0.0871i)T \) |
| 53 | \( 1 + (0.573 + 0.819i)T \) |
| 59 | \( 1 + (-0.996 + 0.0871i)T \) |
| 61 | \( 1 + (-0.642 - 0.766i)T \) |
| 67 | \( 1 + (0.642 - 0.766i)T \) |
| 71 | \( 1 + (-0.173 + 0.984i)T \) |
| 79 | \( 1 + (-0.642 + 0.766i)T \) |
| 83 | \( 1 + (0.707 - 0.707i)T \) |
| 89 | \( 1 + (0.766 + 0.642i)T \) |
| 97 | \( 1 + (0.866 - 0.5i)T \) |
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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
−31.88325645887990768903539555710, −30.11903733782889640884421772474, −29.01735165872380860945549246582, −27.95183004859458276576213289246, −26.72150595270676526152225756255, −26.08333731752029041580895542846, −24.38463687447556359888680862241, −23.86531228315901658715047854676, −22.69187059074762737577574688759, −21.54660319311385163237245558526, −20.77040051658365337175513533162, −18.5666454634284079326105906833, −17.46356982787354727169741797981, −16.63059107249090055350668006424, −16.01255904375447567698584224456, −14.24034846227737106087247289034, −13.43662245961357318907025821559, −11.93340507465931705140260056286, −10.26869200360383660433843663292, −9.25126014282636731382569382841, −7.59195529789751413007912439626, −6.128014774535129492593825676818, −5.19103013013767147390746593697, −3.99186288387699754733042743841, −0.7098013418337631384032358719,
1.53830056324085977662934489009, 2.81179015656353831357605455517, 5.149506708148832745311680191098, 5.87050762631459034305759025157, 7.81875961186345441915996901860, 9.75232537589226257480050248522, 10.57246745577529621380146665928, 11.92584374158121047016139427496, 12.709864244676229927709775738272, 13.94151023039828381301149109388, 15.31220182354905810643618357196, 17.31751300749365924544772957435, 18.20240377662440131329406079057, 18.67886731558557073719923885587, 20.37009662912478619718663677862, 21.61344766015107582784829992654, 22.33755511736744594908772961729, 23.23186152409539811093474055065, 24.62138361602023875293269418952, 25.818148901507773952496188933115, 27.48873835708692952171549439205, 28.312956058776125359977341563863, 29.12569549592030923983931286598, 30.13489890309399417921502861269, 30.80528440539298905219210610978