| L(s) = 1 | + 2-s − 3-s + 4-s + 3·5-s − 6-s + 8-s + 9-s + 3·10-s + 2·11-s − 12-s + 5·13-s − 3·15-s + 16-s + 4·17-s + 18-s − 6·19-s + 3·20-s + 2·22-s + 23-s − 24-s + 4·25-s + 5·26-s − 27-s + 29-s − 3·30-s + 4·31-s + 32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s + 1.34·5-s − 0.408·6-s + 0.353·8-s + 1/3·9-s + 0.948·10-s + 0.603·11-s − 0.288·12-s + 1.38·13-s − 0.774·15-s + 1/4·16-s + 0.970·17-s + 0.235·18-s − 1.37·19-s + 0.670·20-s + 0.426·22-s + 0.208·23-s − 0.204·24-s + 4/5·25-s + 0.980·26-s − 0.192·27-s + 0.185·29-s − 0.547·30-s + 0.718·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6762 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6762 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.203054212\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.203054212\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 5 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - T + p T^{2} \) |
| 41 | \( 1 + T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 12 T + p T^{2} \) |
| 67 | \( 1 + 16 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 6 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 4 T + p T^{2} \) |
| 97 | \( 1 - 3 T + p T^{2} \) |
| 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
−7.918617881851957863498926850459, −6.86968250552726582021652091341, −6.28137344671932013165462585347, −5.96153762451085121171496200820, −5.30976521983415597175653776546, −4.42324800736466071285079030366, −3.73650977170265112461812457567, −2.74464353211888054441904822657, −1.77168877155839281306683984426, −1.07167247769192420833946347945,
1.07167247769192420833946347945, 1.77168877155839281306683984426, 2.74464353211888054441904822657, 3.73650977170265112461812457567, 4.42324800736466071285079030366, 5.30976521983415597175653776546, 5.96153762451085121171496200820, 6.28137344671932013165462585347, 6.86968250552726582021652091341, 7.918617881851957863498926850459
