L(s) = 1 | + (0.916 − 0.400i)2-s + (0.600 + 0.799i)3-s + (0.679 − 0.733i)4-s + (0.751 − 0.660i)5-s + (0.870 + 0.492i)6-s + (0.994 − 0.102i)7-s + (0.328 − 0.944i)8-s + (−0.279 + 0.960i)9-s + (0.423 − 0.905i)10-s + (−0.640 + 0.767i)11-s + (0.994 + 0.102i)12-s + (−0.279 + 0.960i)13-s + (0.870 − 0.492i)14-s + (0.978 + 0.204i)15-s + (−0.0771 − 0.997i)16-s + (−0.376 − 0.926i)17-s + ⋯ |
L(s) = 1 | + (0.916 − 0.400i)2-s + (0.600 + 0.799i)3-s + (0.679 − 0.733i)4-s + (0.751 − 0.660i)5-s + (0.870 + 0.492i)6-s + (0.994 − 0.102i)7-s + (0.328 − 0.944i)8-s + (−0.279 + 0.960i)9-s + (0.423 − 0.905i)10-s + (−0.640 + 0.767i)11-s + (0.994 + 0.102i)12-s + (−0.279 + 0.960i)13-s + (0.870 − 0.492i)14-s + (0.978 + 0.204i)15-s + (−0.0771 − 0.997i)16-s + (−0.376 − 0.926i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 367 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.972 - 0.234i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 367 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.972 - 0.234i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(3.069648882 - 0.3657512542i\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.069648882 - 0.3657512542i\) |
\(L(1)\) |
\(\approx\) |
\(2.315411967 - 0.2173570314i\) |
\(L(1)\) |
\(\approx\) |
\(2.315411967 - 0.2173570314i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 367 | \( 1 \) |
good | 2 | \( 1 + (0.916 - 0.400i)T \) |
| 3 | \( 1 + (0.600 + 0.799i)T \) |
| 5 | \( 1 + (0.751 - 0.660i)T \) |
| 7 | \( 1 + (0.994 - 0.102i)T \) |
| 11 | \( 1 + (-0.640 + 0.767i)T \) |
| 13 | \( 1 + (-0.279 + 0.960i)T \) |
| 17 | \( 1 + (-0.376 - 0.926i)T \) |
| 19 | \( 1 + (-0.967 - 0.254i)T \) |
| 23 | \( 1 + (-0.0771 + 0.997i)T \) |
| 29 | \( 1 + (-0.988 - 0.153i)T \) |
| 31 | \( 1 + (0.679 - 0.733i)T \) |
| 37 | \( 1 + (-0.469 + 0.882i)T \) |
| 41 | \( 1 + (-0.640 - 0.767i)T \) |
| 43 | \( 1 + (-0.967 + 0.254i)T \) |
| 47 | \( 1 + (-0.998 + 0.0514i)T \) |
| 53 | \( 1 + (0.751 - 0.660i)T \) |
| 59 | \( 1 + (0.229 + 0.973i)T \) |
| 61 | \( 1 + (0.870 - 0.492i)T \) |
| 67 | \( 1 + (0.229 - 0.973i)T \) |
| 71 | \( 1 + (0.679 + 0.733i)T \) |
| 73 | \( 1 + (0.952 + 0.304i)T \) |
| 79 | \( 1 + (0.978 - 0.204i)T \) |
| 83 | \( 1 + T \) |
| 89 | \( 1 + (-0.998 + 0.0514i)T \) |
| 97 | \( 1 + (-0.0771 + 0.997i)T \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−24.723482404088707867352058786484, −23.92841009928157479389673960018, −23.14633086241144346954056582048, −22.04053654415482045488669435279, −21.23664456879524103041959235820, −20.628346714609797341644903748223, −19.48515494632089299211842981148, −18.358065062863358219699502355158, −17.68347386408687491636478111579, −16.8228063472630119787286157597, −15.19639365321912970289877634204, −14.79641902288796421980173212303, −13.969961080145946748521204827290, −13.16464917419846034574225505753, −12.479698553677880051710950425658, −11.17997289959856094493707712730, −10.4148083860703951724644210559, −8.516354951181312368433161515832, −8.07390103283371072784150301670, −6.903078343131878129086616810710, −6.05779623372738097641699374497, −5.16823474546150810499703823869, −3.63340946579518680054296574453, −2.58908443258824853263536777842, −1.82377554858344950064818145518,
1.83649237053710859427637180480, 2.3568169853456862037597176382, 3.96873114170460928292247312889, 4.86132466712191258583061849118, 5.25442149074055210527969703411, 6.853190382608665748631050674586, 8.13349313206914175123783979047, 9.36199004765378845041192169887, 10.00515913477005766935650207859, 11.08645422106783926294302687701, 11.94902850337561507593777477283, 13.32897137492326151891590904919, 13.71444053649024993661281424474, 14.77092148078260039761102342665, 15.37576748012647087927902333532, 16.4755669419321867614328731605, 17.37478929246609891334173523265, 18.66720948119339475859078417495, 19.85432757836674812108336639101, 20.60958231092648258604674447955, 21.110749371753045020634357935713, 21.676663881057843615769590881535, 22.694149393109882019163915734465, 23.864815219721125651279816447544, 24.47595850062411285622597881730