L(s) = 1 | − 2-s + 3-s − 4-s − 4·5-s − 6-s + 7-s + 3·8-s + 9-s + 4·10-s − 4·11-s − 12-s + 13-s − 14-s − 4·15-s − 16-s + 17-s − 18-s + 4·20-s + 21-s + 4·22-s − 4·23-s + 3·24-s + 11·25-s − 26-s + 27-s − 28-s − 4·29-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s − 1/2·4-s − 1.78·5-s − 0.408·6-s + 0.377·7-s + 1.06·8-s + 1/3·9-s + 1.26·10-s − 1.20·11-s − 0.288·12-s + 0.277·13-s − 0.267·14-s − 1.03·15-s − 1/4·16-s + 0.242·17-s − 0.235·18-s + 0.894·20-s + 0.218·21-s + 0.852·22-s − 0.834·23-s + 0.612·24-s + 11/5·25-s − 0.196·26-s + 0.192·27-s − 0.188·28-s − 0.742·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4641 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4641 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 - T \) |
good | 2 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 2 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - 10 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 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
−7.911207553416769168909674000361, −7.73505906393945326199088403431, −6.98064826192607076362513149655, −5.63323242695575225195739109211, −4.68430646450251969128276399415, −4.18388615288972438683834154897, −3.44663436767775664662824838979, −2.43707217822062715094038298163, −1.03664910738811434228075373099, 0,
1.03664910738811434228075373099, 2.43707217822062715094038298163, 3.44663436767775664662824838979, 4.18388615288972438683834154897, 4.68430646450251969128276399415, 5.63323242695575225195739109211, 6.98064826192607076362513149655, 7.73505906393945326199088403431, 7.911207553416769168909674000361