L(s) = 1 | + 2-s + 3-s + 4-s − 4·5-s + 6-s + 2·7-s + 8-s + 9-s − 4·10-s − 11-s + 12-s + 2·14-s − 4·15-s + 16-s + 17-s + 18-s + 19-s − 4·20-s + 2·21-s − 22-s + 7·23-s + 24-s + 11·25-s + 27-s + 2·28-s − 29-s − 4·30-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s − 1.78·5-s + 0.408·6-s + 0.755·7-s + 0.353·8-s + 1/3·9-s − 1.26·10-s − 0.301·11-s + 0.288·12-s + 0.534·14-s − 1.03·15-s + 1/4·16-s + 0.242·17-s + 0.235·18-s + 0.229·19-s − 0.894·20-s + 0.436·21-s − 0.213·22-s + 1.45·23-s + 0.204·24-s + 11/5·25-s + 0.192·27-s + 0.377·28-s − 0.185·29-s − 0.730·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 189618 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189618 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.516150288\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.516150288\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 \) |
| 17 | \( 1 - T \) |
good | 5 | \( 1 + 4 T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 - 7 T + p T^{2} \) |
| 29 | \( 1 + T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 3 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 7 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 + T + p T^{2} \) |
| 97 | \( 1 - 7 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.10401770427002, −12.62248660192897, −12.13984274561870, −11.76364664391964, −11.37451747505108, −10.89859010320566, −10.49956437406420, −9.923220044791918, −9.065836113053528, −8.767062942996767, −8.128576257512179, −7.851161277234023, −7.452208861396177, −6.932055443916562, −6.496580837650973, −5.649723917436906, −4.953279502320924, −4.702299091403668, −4.280624415903143, −3.521107596840553, −3.250550269133491, −2.749476841014114, −1.949407704104954, −1.171783225545526, −0.5606047909444037,
0.5606047909444037, 1.171783225545526, 1.949407704104954, 2.749476841014114, 3.250550269133491, 3.521107596840553, 4.280624415903143, 4.702299091403668, 4.953279502320924, 5.649723917436906, 6.496580837650973, 6.932055443916562, 7.452208861396177, 7.851161277234023, 8.128576257512179, 8.767062942996767, 9.065836113053528, 9.923220044791918, 10.49956437406420, 10.89859010320566, 11.37451747505108, 11.76364664391964, 12.13984274561870, 12.62248660192897, 13.10401770427002