L(s) = 1 | + 2-s − 3-s + 4-s + 5-s − 6-s + 7-s + 8-s − 2·9-s + 10-s − 3·11-s − 12-s − 6·13-s + 14-s − 15-s + 16-s − 2·17-s − 2·18-s + 20-s − 21-s − 3·22-s − 6·23-s − 24-s − 4·25-s − 6·26-s + 5·27-s + 28-s − 30-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s + 0.377·7-s + 0.353·8-s − 2/3·9-s + 0.316·10-s − 0.904·11-s − 0.288·12-s − 1.66·13-s + 0.267·14-s − 0.258·15-s + 1/4·16-s − 0.485·17-s − 0.471·18-s + 0.223·20-s − 0.218·21-s − 0.639·22-s − 1.25·23-s − 0.204·24-s − 4/5·25-s − 1.17·26-s + 0.962·27-s + 0.188·28-s − 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1526 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1526 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 109 | \( 1 + T \) |
good | 3 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 - 14 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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
−9.117828627426357441137464936094, −8.065281149853209060616012078115, −7.37254497052080588577371673056, −6.39940323385614452260114207042, −5.54350958210853846360207828376, −5.11031490968013398287456926487, −4.17175621228422883644320076210, −2.76243985087989233558008934289, −2.06208988293120625162340121374, 0,
2.06208988293120625162340121374, 2.76243985087989233558008934289, 4.17175621228422883644320076210, 5.11031490968013398287456926487, 5.54350958210853846360207828376, 6.39940323385614452260114207042, 7.37254497052080588577371673056, 8.065281149853209060616012078115, 9.117828627426357441137464936094