L(s) = 1 | + (0.743 − 0.669i)2-s + (0.207 + 0.978i)3-s + (0.104 − 0.994i)4-s + (0.809 + 0.587i)6-s + (−0.933 − 0.358i)7-s + (−0.587 − 0.809i)8-s + (−0.913 + 0.406i)9-s + (0.0523 − 0.998i)11-s + (0.994 − 0.104i)12-s + (−0.933 + 0.358i)13-s + (−0.933 + 0.358i)14-s + (−0.978 − 0.207i)16-s + (−0.707 + 0.707i)17-s + (−0.406 + 0.913i)18-s + (0.629 − 0.777i)19-s + ⋯ |
L(s) = 1 | + (0.743 − 0.669i)2-s + (0.207 + 0.978i)3-s + (0.104 − 0.994i)4-s + (0.809 + 0.587i)6-s + (−0.933 − 0.358i)7-s + (−0.587 − 0.809i)8-s + (−0.913 + 0.406i)9-s + (0.0523 − 0.998i)11-s + (0.994 − 0.104i)12-s + (−0.933 + 0.358i)13-s + (−0.933 + 0.358i)14-s + (−0.978 − 0.207i)16-s + (−0.707 + 0.707i)17-s + (−0.406 + 0.913i)18-s + (0.629 − 0.777i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.573 + 0.818i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.573 + 0.818i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7792869540 + 0.4054317835i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7792869540 + 0.4054317835i\) |
\(L(1)\) |
\(\approx\) |
\(1.084732314 - 0.2496433474i\) |
\(L(1)\) |
\(\approx\) |
\(1.084732314 - 0.2496433474i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 241 | \( 1 \) |
good | 2 | \( 1 + (0.743 - 0.669i)T \) |
| 3 | \( 1 + (0.207 + 0.978i)T \) |
| 7 | \( 1 + (-0.933 - 0.358i)T \) |
| 11 | \( 1 + (0.0523 - 0.998i)T \) |
| 13 | \( 1 + (-0.933 + 0.358i)T \) |
| 17 | \( 1 + (-0.707 + 0.707i)T \) |
| 19 | \( 1 + (0.629 - 0.777i)T \) |
| 23 | \( 1 + (-0.707 - 0.707i)T \) |
| 29 | \( 1 + (-0.866 - 0.5i)T \) |
| 31 | \( 1 + (-0.258 - 0.965i)T \) |
| 37 | \( 1 + (0.965 - 0.258i)T \) |
| 41 | \( 1 + (-0.951 - 0.309i)T \) |
| 43 | \( 1 + (-0.156 + 0.987i)T \) |
| 47 | \( 1 + iT \) |
| 53 | \( 1 + (-0.866 + 0.5i)T \) |
| 59 | \( 1 + (0.866 + 0.5i)T \) |
| 61 | \( 1 + iT \) |
| 67 | \( 1 + (0.866 + 0.5i)T \) |
| 71 | \( 1 + (-0.258 - 0.965i)T \) |
| 73 | \( 1 + (0.156 + 0.987i)T \) |
| 79 | \( 1 + (0.587 + 0.809i)T \) |
| 83 | \( 1 + (0.978 + 0.207i)T \) |
| 89 | \( 1 + (-0.998 - 0.0523i)T \) |
| 97 | \( 1 + (0.5 + 0.866i)T \) |
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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
−17.6235350435576246858371545308, −16.90048402066068204295708258294, −16.26722286041642372759711948307, −15.47234516765799042227115191649, −14.99471109391289691143005028947, −14.24509486951647064798793646024, −13.71855292987755862217493586710, −12.83807563124768734242695888443, −12.69753844292280414262919977009, −11.88188455183849837401324693298, −11.535499341796758852485856118865, −10.147482215980251246618555724193, −9.46440996431238698420800080575, −8.821282735344820217786662315995, −7.8877771250365279219584937459, −7.39157057249096220487939765057, −6.81621049761349589284270660969, −6.28530501256259928405054443744, −5.34675182534089175104008897296, −4.98465806092583939803516144503, −3.68390059829363593371815803608, −3.25448364649014470992931235713, −2.31929044702353706244943114182, −1.83731400368134277991078946141, −0.19853079585444988997807016676,
0.73776244446778476619415258722, 2.15592200471227105526988742735, 2.72393381526873853655443346460, 3.410777679402252762449418292643, 4.1418175113873311244011095608, 4.52397968827002012823864058565, 5.54484431683088634860352519229, 6.05949494269449977271834459097, 6.77149399638020914516709695134, 7.79749776484885662523605282314, 8.78201042935108240422145164731, 9.50905990476133851419094498066, 9.79406563912435361914733232796, 10.62816619392452301335143452738, 11.19664417312637945322218298917, 11.70177582687597127615734232180, 12.67163125374008456824228344809, 13.2741091837363683198672188024, 13.84283270495581953795546635363, 14.47658006449624261830123706072, 15.142323929693236345163557099591, 15.70603286847958055678369904846, 16.44660802683076513922494578301, 16.821352509851162452645193357434, 17.814736089053583812635271635210