L(s) = 1 | + (−0.0172 + 0.999i)2-s + (−0.900 + 0.433i)3-s + (−0.999 − 0.0345i)4-s + (0.0517 + 0.998i)5-s + (−0.418 − 0.908i)6-s + (0.978 + 0.205i)7-s + (0.0517 − 0.998i)8-s + (0.623 − 0.781i)9-s + (−0.999 + 0.0345i)10-s + (−0.354 − 0.935i)11-s + (0.915 − 0.402i)12-s + (−0.222 + 0.974i)13-s + (−0.222 + 0.974i)14-s + (−0.479 − 0.877i)15-s + (0.997 + 0.0689i)16-s + (0.509 + 0.860i)17-s + ⋯ |
L(s) = 1 | + (−0.0172 + 0.999i)2-s + (−0.900 + 0.433i)3-s + (−0.999 − 0.0345i)4-s + (0.0517 + 0.998i)5-s + (−0.418 − 0.908i)6-s + (0.978 + 0.205i)7-s + (0.0517 − 0.998i)8-s + (0.623 − 0.781i)9-s + (−0.999 + 0.0345i)10-s + (−0.354 − 0.935i)11-s + (0.915 − 0.402i)12-s + (−0.222 + 0.974i)13-s + (−0.222 + 0.974i)14-s + (−0.479 − 0.877i)15-s + (0.997 + 0.0689i)16-s + (0.509 + 0.860i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.990 - 0.138i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.990 - 0.138i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(-0.06147426077 + 0.8830315852i\) |
\(L(\frac12)\) |
\(\approx\) |
\(-0.06147426077 + 0.8830315852i\) |
\(L(1)\) |
\(\approx\) |
\(0.4897473067 + 0.6316570674i\) |
\(L(1)\) |
\(\approx\) |
\(0.4897473067 + 0.6316570674i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 547 | \( 1 \) |
good | 2 | \( 1 + (-0.0172 + 0.999i)T \) |
| 3 | \( 1 + (-0.900 + 0.433i)T \) |
| 5 | \( 1 + (0.0517 + 0.998i)T \) |
| 7 | \( 1 + (0.978 + 0.205i)T \) |
| 11 | \( 1 + (-0.354 - 0.935i)T \) |
| 13 | \( 1 + (-0.222 + 0.974i)T \) |
| 17 | \( 1 + (0.509 + 0.860i)T \) |
| 19 | \( 1 + (0.675 + 0.736i)T \) |
| 23 | \( 1 + (0.940 + 0.338i)T \) |
| 29 | \( 1 + (-0.700 + 0.713i)T \) |
| 31 | \( 1 + (-0.868 - 0.495i)T \) |
| 37 | \( 1 + (0.851 + 0.524i)T \) |
| 41 | \( 1 + T \) |
| 43 | \( 1 + (-0.596 - 0.802i)T \) |
| 47 | \( 1 + (0.120 + 0.992i)T \) |
| 53 | \( 1 + (-0.479 - 0.877i)T \) |
| 59 | \( 1 + (0.568 + 0.822i)T \) |
| 61 | \( 1 + (-0.792 + 0.609i)T \) |
| 67 | \( 1 + (0.915 - 0.402i)T \) |
| 71 | \( 1 + (-0.289 - 0.957i)T \) |
| 73 | \( 1 + (-0.479 + 0.877i)T \) |
| 79 | \( 1 + (-0.596 - 0.802i)T \) |
| 83 | \( 1 + (-0.748 + 0.663i)T \) |
| 89 | \( 1 + (-0.792 - 0.609i)T \) |
| 97 | \( 1 + (-0.832 - 0.553i)T \) |
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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
−23.10463010671134069091755248604, −22.03514758472860380806875375002, −21.1587975341613699583426580231, −20.41494049175940302079220317879, −19.880280285188649122376528453843, −18.571657960823355812547151290762, −17.88031587517419852840584039018, −17.38551405584263985988705791252, −16.5408480702616047571043173887, −15.303767368914045133983631330780, −14.11347498610881062374385935234, −13.01612148572846841094314156941, −12.69096351525430476473391943207, −11.69298424136816840847713726229, −11.10386915800000881914797220718, −10.07778523685309983819426139799, −9.237128402441988465102737806420, −7.96131182516040247229580514511, −7.37901937299288309616990347637, −5.43703875287914803786809652911, −5.09186645125120471663461414833, −4.28540639107046482394149912953, −2.58353453178418880864141326643, −1.47415454206191334537758730041, −0.611755063926397053945562119526,
1.42132582402325381230428419440, 3.36351658540923325450480813488, 4.30899926810195553305014145510, 5.48393317251241885992076717760, 5.92745472913104506951694435606, 7.04416928010908124279888148060, 7.79920257765417323544775249667, 8.98511688905031264896037672378, 9.9556721183363104707595281653, 10.93773793334163123497164890829, 11.54150283495826794039255334291, 12.76453580265848047851008228713, 13.990024911741361208695984627085, 14.67187107591052391324223195022, 15.278836519779446602725986172323, 16.392821712089136945683109352563, 16.898704190593143206499461333045, 17.86954156722609581090566641273, 18.52870052348177340470540794189, 19.0721898110315882265216942429, 20.96216920284569169757099442028, 21.65205874065450144962795738672, 22.15511836784372434206153823046, 23.130475719529901465190249778877, 23.86433517464229654281197827346