L(s) = 1 | + (−0.703 + 0.710i)2-s + (−0.893 + 0.449i)3-s + (−0.0101 − 0.999i)4-s + (−0.784 − 0.620i)5-s + (0.309 − 0.951i)6-s + (−0.250 + 0.968i)7-s + (0.717 + 0.696i)8-s + (0.595 − 0.803i)9-s + (0.992 − 0.121i)10-s + (−0.758 + 0.651i)11-s + (0.458 + 0.888i)12-s + (−0.954 + 0.299i)13-s + (−0.511 − 0.859i)14-s + (0.979 + 0.201i)15-s + (−0.999 + 0.0202i)16-s + (−0.731 + 0.681i)17-s + ⋯ |
L(s) = 1 | + (−0.703 + 0.710i)2-s + (−0.893 + 0.449i)3-s + (−0.0101 − 0.999i)4-s + (−0.784 − 0.620i)5-s + (0.309 − 0.951i)6-s + (−0.250 + 0.968i)7-s + (0.717 + 0.696i)8-s + (0.595 − 0.803i)9-s + (0.992 − 0.121i)10-s + (−0.758 + 0.651i)11-s + (0.458 + 0.888i)12-s + (−0.954 + 0.299i)13-s + (−0.511 − 0.859i)14-s + (0.979 + 0.201i)15-s + (−0.999 + 0.0202i)16-s + (−0.731 + 0.681i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 311 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.970 - 0.241i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 311 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.970 - 0.241i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2871070718 - 0.03512063171i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2871070718 - 0.03512063171i\) |
\(L(1)\) |
\(\approx\) |
\(0.3903721420 + 0.1279038078i\) |
\(L(1)\) |
\(\approx\) |
\(0.3903721420 + 0.1279038078i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 311 | \( 1 \) |
good | 2 | \( 1 + (-0.703 + 0.710i)T \) |
| 3 | \( 1 + (-0.893 + 0.449i)T \) |
| 5 | \( 1 + (-0.784 - 0.620i)T \) |
| 7 | \( 1 + (-0.250 + 0.968i)T \) |
| 11 | \( 1 + (-0.758 + 0.651i)T \) |
| 13 | \( 1 + (-0.954 + 0.299i)T \) |
| 17 | \( 1 + (-0.731 + 0.681i)T \) |
| 19 | \( 1 + (0.270 - 0.962i)T \) |
| 23 | \( 1 + (-0.171 - 0.985i)T \) |
| 29 | \( 1 + (-0.0910 - 0.995i)T \) |
| 31 | \( 1 + (0.992 + 0.121i)T \) |
| 37 | \( 1 + (0.843 - 0.537i)T \) |
| 41 | \( 1 + (-0.0506 + 0.998i)T \) |
| 43 | \( 1 + (-0.366 - 0.930i)T \) |
| 47 | \( 1 + (0.918 - 0.394i)T \) |
| 53 | \( 1 + (-0.998 + 0.0607i)T \) |
| 59 | \( 1 + (0.771 - 0.635i)T \) |
| 61 | \( 1 + (0.347 + 0.937i)T \) |
| 67 | \( 1 + (0.934 - 0.356i)T \) |
| 71 | \( 1 + (-0.983 - 0.181i)T \) |
| 73 | \( 1 + (0.960 + 0.279i)T \) |
| 79 | \( 1 + (0.628 - 0.778i)T \) |
| 83 | \( 1 + (0.979 - 0.201i)T \) |
| 89 | \( 1 + (-0.250 - 0.968i)T \) |
| 97 | \( 1 + (0.902 - 0.431i)T \) |
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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
−25.43172295194185310445284379263, −24.217733914399773582120733846, −23.35689957462799817196355636363, −22.51384392133254529018836696519, −21.88735364246613338579003495242, −20.57396065334340271124587359505, −19.65817917187319220937041507470, −18.9669311774615873484606217331, −18.15582853758959267445474989989, −17.33911872127113411352773095531, −16.38780756613780924614551757669, −15.72221528306355969654937457830, −13.9748550612105681620701031325, −13.0412310730800439014150100515, −12.07605137023501950892933106703, −11.24680064425693230637198506767, −10.57738881282198699941991651243, −9.77816410442218331292407627570, −7.984619849865743703253403872002, −7.49346444469365890046092501081, −6.57475565322646006004352915641, −4.915395781443203023634864349819, −3.69672016595565258773760733482, −2.562458530673995116175509094661, −0.88503679628898914088493897090,
0.36317780594895974769408707729, 2.28857119927678176384721172170, 4.47121783466572577113185094580, 5.01579979307767614170643888308, 6.15666322391228945101333010233, 7.160693074976354533439889887246, 8.30177780403633952617493398419, 9.26582348714815918411755361425, 10.09200754517246185527860740181, 11.24316721718271758648106265525, 12.11243357772010514496466471770, 13.04532663712384192758177447641, 14.93007508177684603414513656138, 15.43292988083617623178431491783, 16.07224230783517558201878534310, 17.03176747060743917001263568550, 17.77001118972749147776241118736, 18.72805552414150677575913579614, 19.62207588847957846200012918891, 20.61435014491181437833731731097, 21.84973203044899392961207614062, 22.70099524662751116679302023384, 23.63947646272901416186446263770, 24.25856945605178818279245208277, 25.072565538757462490726084735025