L(s) = 1 | − 2-s + 4-s − 5-s − 8-s + 10-s − 2·11-s + 2·13-s + 16-s − 4·17-s − 20-s + 2·22-s − 23-s + 25-s − 2·26-s + 8·29-s − 2·31-s − 32-s + 4·34-s − 8·37-s + 40-s + 10·41-s − 2·43-s − 2·44-s + 46-s + 6·47-s − 50-s + 2·52-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.353·8-s + 0.316·10-s − 0.603·11-s + 0.554·13-s + 1/4·16-s − 0.970·17-s − 0.223·20-s + 0.426·22-s − 0.208·23-s + 1/5·25-s − 0.392·26-s + 1.48·29-s − 0.359·31-s − 0.176·32-s + 0.685·34-s − 1.31·37-s + 0.158·40-s + 1.56·41-s − 0.304·43-s − 0.301·44-s + 0.147·46-s + 0.875·47-s − 0.141·50-s + 0.277·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 101430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 101430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| 23 | \( 1 + T \) |
good | 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.99769242707891, −13.54963796700903, −12.81445011906085, −12.56872559862485, −11.94721077117171, −11.40323991592816, −11.01821526254366, −10.47424433348609, −10.21927414982092, −9.415736080357626, −9.015541305005169, −8.398844682494101, −8.227886959704803, −7.411051505084003, −7.164004582096342, −6.370055782487563, −6.112756689715690, −5.267819954538057, −4.764889547820226, −4.089111086551331, −3.529464835892341, −2.779602310716887, −2.314979924699066, −1.517400570337017, −0.7557520455751736, 0,
0.7557520455751736, 1.517400570337017, 2.314979924699066, 2.779602310716887, 3.529464835892341, 4.089111086551331, 4.764889547820226, 5.267819954538057, 6.112756689715690, 6.370055782487563, 7.164004582096342, 7.411051505084003, 8.227886959704803, 8.398844682494101, 9.015541305005169, 9.415736080357626, 10.21927414982092, 10.47424433348609, 11.01821526254366, 11.40323991592816, 11.94721077117171, 12.56872559862485, 12.81445011906085, 13.54963796700903, 13.99769242707891