L(s) = 1 | + (−0.913 − 0.406i)2-s + (−0.669 + 0.743i)3-s + (0.669 + 0.743i)4-s + (0.913 − 0.406i)6-s + (−0.309 + 0.951i)7-s + (−0.309 − 0.951i)8-s + (−0.104 − 0.994i)9-s − 12-s + (0.104 + 0.994i)13-s + (0.669 − 0.743i)14-s + (−0.104 + 0.994i)16-s + (0.104 − 0.994i)17-s + (−0.309 + 0.951i)18-s + (−0.5 − 0.866i)21-s + (0.5 − 0.866i)23-s + (0.913 + 0.406i)24-s + ⋯ |
L(s) = 1 | + (−0.913 − 0.406i)2-s + (−0.669 + 0.743i)3-s + (0.669 + 0.743i)4-s + (0.913 − 0.406i)6-s + (−0.309 + 0.951i)7-s + (−0.309 − 0.951i)8-s + (−0.104 − 0.994i)9-s − 12-s + (0.104 + 0.994i)13-s + (0.669 − 0.743i)14-s + (−0.104 + 0.994i)16-s + (0.104 − 0.994i)17-s + (−0.309 + 0.951i)18-s + (−0.5 − 0.866i)21-s + (0.5 − 0.866i)23-s + (0.913 + 0.406i)24-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.367 + 0.929i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.367 + 0.929i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3254316131 + 0.4786222764i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3254316131 + 0.4786222764i\) |
\(L(1)\) |
\(\approx\) |
\(0.5327251155 + 0.1619627366i\) |
\(L(1)\) |
\(\approx\) |
\(0.5327251155 + 0.1619627366i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 11 | \( 1 \) |
| 19 | \( 1 \) |
good | 2 | \( 1 + (-0.913 - 0.406i)T \) |
| 3 | \( 1 + (-0.669 + 0.743i)T \) |
| 7 | \( 1 + (-0.309 + 0.951i)T \) |
| 13 | \( 1 + (0.104 + 0.994i)T \) |
| 17 | \( 1 + (0.104 - 0.994i)T \) |
| 23 | \( 1 + (0.5 - 0.866i)T \) |
| 29 | \( 1 + (0.669 + 0.743i)T \) |
| 31 | \( 1 + (-0.809 + 0.587i)T \) |
| 37 | \( 1 + (-0.309 + 0.951i)T \) |
| 41 | \( 1 + (0.669 - 0.743i)T \) |
| 43 | \( 1 + (0.5 + 0.866i)T \) |
| 47 | \( 1 + (0.978 + 0.207i)T \) |
| 53 | \( 1 + (0.104 + 0.994i)T \) |
| 59 | \( 1 + (-0.978 + 0.207i)T \) |
| 61 | \( 1 + (0.913 - 0.406i)T \) |
| 67 | \( 1 + (0.5 - 0.866i)T \) |
| 71 | \( 1 + (-0.104 + 0.994i)T \) |
| 73 | \( 1 + (0.978 - 0.207i)T \) |
| 79 | \( 1 + (0.913 + 0.406i)T \) |
| 83 | \( 1 + (0.809 + 0.587i)T \) |
| 89 | \( 1 + (-0.5 + 0.866i)T \) |
| 97 | \( 1 + (-0.913 - 0.406i)T \) |
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
−21.1498559043222966173766910241, −20.16426970118919753702460790821, −19.50536930299862772498951948215, −18.98447002408466519528044255283, −17.93367502047533401350943314055, −17.440473185891822393751473844006, −16.882182120774116394363595231590, −16.07321900662639357184139115986, −15.25082787118605323855034410797, −14.2282266139100252082371777095, −13.31730925326901109711456340409, −12.60428152670888658692868295857, −11.53178690306137284604794290772, −10.7066396031508703452714786847, −10.29214113494351269269778482618, −9.200539006312648934121342281910, −8.007020823100449489079182308580, −7.595729918084366170668911734601, −6.7286640961441604374714410136, −5.947399458241295926572679125216, −5.2191536813369509407031887730, −3.795715851683234527285099731015, −2.424829742177262428284824260305, −1.316715171158199425275962285745, −0.45439393155846958877973368519,
1.06135274452310523229652074087, 2.42496410169269953010714995370, 3.245925538158762600393008403369, 4.36527849997990242781696045198, 5.355299541234550903082678856787, 6.43621095317912116965317369150, 7.0287568841398752801103923572, 8.42316516222185761837660673985, 9.20829413360428488319378367884, 9.55743134728509365012865044558, 10.69831993547985242114562087518, 11.2161675498999287632310256017, 12.2232105024843880433985208496, 12.44111361076345214238306911089, 13.98217263241020971419713276778, 15.04661332217673286475707625305, 15.88240154712851921725779425818, 16.33614763194464735537599154546, 17.05357324789781469043474660294, 18.04715572794761692247371898868, 18.521861402315949758635795172576, 19.30553521537878615689684431938, 20.34826508677395020934649032067, 20.990113780155602927501486417757, 21.71692011411586553662891291349