L(s) = 1 | + (0.951 + 0.309i)2-s + (−0.233 − 0.972i)3-s + (0.809 + 0.587i)4-s + (0.587 − 0.809i)5-s + (0.0784 − 0.996i)6-s + (0.0784 + 0.996i)7-s + (0.587 + 0.809i)8-s + (−0.891 + 0.453i)9-s + (0.809 − 0.587i)10-s + (0.996 + 0.0784i)11-s + (0.382 − 0.923i)12-s + (−0.382 + 0.923i)13-s + (−0.233 + 0.972i)14-s + (−0.923 − 0.382i)15-s + (0.309 + 0.951i)16-s + (−0.649 − 0.760i)17-s + ⋯ |
L(s) = 1 | + (0.951 + 0.309i)2-s + (−0.233 − 0.972i)3-s + (0.809 + 0.587i)4-s + (0.587 − 0.809i)5-s + (0.0784 − 0.996i)6-s + (0.0784 + 0.996i)7-s + (0.587 + 0.809i)8-s + (−0.891 + 0.453i)9-s + (0.809 − 0.587i)10-s + (0.996 + 0.0784i)11-s + (0.382 − 0.923i)12-s + (−0.382 + 0.923i)13-s + (−0.233 + 0.972i)14-s + (−0.923 − 0.382i)15-s + (0.309 + 0.951i)16-s + (−0.649 − 0.760i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1601 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0635i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1601 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0635i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(3.233003149 - 0.1028803086i\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.233003149 - 0.1028803086i\) |
\(L(1)\) |
\(\approx\) |
\(2.001365064 - 0.08781894441i\) |
\(L(1)\) |
\(\approx\) |
\(2.001365064 - 0.08781894441i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 1601 | \( 1 \) |
good | 2 | \( 1 + (0.951 + 0.309i)T \) |
| 3 | \( 1 + (-0.233 - 0.972i)T \) |
| 5 | \( 1 + (0.587 - 0.809i)T \) |
| 7 | \( 1 + (0.0784 + 0.996i)T \) |
| 11 | \( 1 + (0.996 + 0.0784i)T \) |
| 13 | \( 1 + (-0.382 + 0.923i)T \) |
| 17 | \( 1 + (-0.649 - 0.760i)T \) |
| 19 | \( 1 + (0.309 - 0.951i)T \) |
| 23 | \( 1 + (-0.233 + 0.972i)T \) |
| 29 | \( 1 + (0.809 - 0.587i)T \) |
| 31 | \( 1 + (-0.809 - 0.587i)T \) |
| 37 | \( 1 + (0.987 + 0.156i)T \) |
| 41 | \( 1 + (0.987 + 0.156i)T \) |
| 43 | \( 1 + (0.707 - 0.707i)T \) |
| 47 | \( 1 + (0.987 + 0.156i)T \) |
| 53 | \( 1 + (-0.809 + 0.587i)T \) |
| 59 | \( 1 + (0.996 + 0.0784i)T \) |
| 61 | \( 1 + (0.309 + 0.951i)T \) |
| 67 | \( 1 + (-0.707 + 0.707i)T \) |
| 71 | \( 1 + (0.972 - 0.233i)T \) |
| 73 | \( 1 + (0.233 - 0.972i)T \) |
| 79 | \( 1 + T \) |
| 83 | \( 1 + (0.923 + 0.382i)T \) |
| 89 | \( 1 + (-0.987 - 0.156i)T \) |
| 97 | \( 1 + (-0.156 + 0.987i)T \) |
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
−20.59104506857634973872826073822, −19.96191526635003870877543143770, −19.414558653970446064928771929567, −18.10298876276010090641915490967, −17.36745855296700965285439864807, −16.63543934408370706943691850483, −15.921678167152743331802637910, −14.915194199285782307325507187297, −14.38341267726858770670998271865, −14.10333200461763321708565588092, −12.92672143382689572555001113343, −12.21551323143214963218884311575, −10.98971211378511203840975119085, −10.838140989338688482442609028765, −10.07003346491128361828669037460, −9.441783991014190211669704814519, −8.07329148151910104676026152835, −6.934292960386335553058661401285, −6.27973716338836456385433702937, −5.59553119406750925291123719990, −4.58198374679910191452071028319, −3.8631539502397642213322687710, −3.256402874876681441959830712812, −2.25838726397811654337686502661, −1.02222476940342786661202218290,
1.14538481232127215076965422283, 2.195034558402357982130233474973, 2.57198151576627021023208903499, 4.13176805504359773473716382382, 4.93166398557898492212986019326, 5.69573481518694233412624083174, 6.31860105508870866722729921866, 7.06418086386191241190375485627, 7.92385864459856784109843867391, 9.04101642264934455880358056524, 9.33166996285857349268815675803, 11.13776614779676265197286126178, 11.768849157981917068099854367276, 12.156862935765466667503547465061, 12.99020123994340601083523100820, 13.694005122839035226291550541013, 14.16574941161412656534411323397, 15.11418873338925708707728624027, 16.03996054126010532748790833618, 16.68516362862081204335672973844, 17.54217752574776625070322021788, 17.89465306789884462410094033445, 19.16560134781068623451588170369, 19.75178901536313060940421704650, 20.51004015532985623316304168240