L(s) = 1 | − 2-s − 3-s + 4-s + 6-s + 7-s − 8-s + 9-s − 4·11-s − 12-s − 2·13-s − 14-s + 16-s + 6·17-s − 18-s − 21-s + 4·22-s + 8·23-s + 24-s + 2·26-s − 27-s + 28-s − 10·29-s + 8·31-s − 32-s + 4·33-s − 6·34-s + 36-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s − 1.20·11-s − 0.288·12-s − 0.554·13-s − 0.267·14-s + 1/4·16-s + 1.45·17-s − 0.235·18-s − 0.218·21-s + 0.852·22-s + 1.66·23-s + 0.204·24-s + 0.392·26-s − 0.192·27-s + 0.188·28-s − 1.85·29-s + 1.43·31-s − 0.176·32-s + 0.696·33-s − 1.02·34-s + 1/6·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 379050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 379050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| 19 | \( 1 \) |
good | 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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
−12.63217708033879, −12.20559449253725, −11.71995736377392, −11.15450309019835, −11.00120716090249, −10.43168586844661, −9.926374985463192, −9.727770619080022, −9.161282096068201, −8.546149745406711, −8.073171239962601, −7.646205626831047, −7.383139309240766, −6.797257576998471, −6.316767144990379, −5.595990042406940, −5.294670282848913, −4.985845680866745, −4.331235734187865, −3.542101436677911, −3.068327016996871, −2.525628242165059, −1.922561681020966, −1.217914488132770, −0.7266399115984335, 0,
0.7266399115984335, 1.217914488132770, 1.922561681020966, 2.525628242165059, 3.068327016996871, 3.542101436677911, 4.331235734187865, 4.985845680866745, 5.294670282848913, 5.595990042406940, 6.316767144990379, 6.797257576998471, 7.383139309240766, 7.646205626831047, 8.073171239962601, 8.546149745406711, 9.161282096068201, 9.727770619080022, 9.926374985463192, 10.43168586844661, 11.00120716090249, 11.15450309019835, 11.71995736377392, 12.20559449253725, 12.63217708033879