L(s) = 1 | + 2-s + 3-s + 4-s − 5-s + 6-s + 3.91·7-s + 8-s + 9-s − 10-s + 11-s + 12-s + 5.91·13-s + 3.91·14-s − 15-s + 16-s − 17-s + 18-s + 5.91·19-s − 20-s + 3.91·21-s + 22-s − 1.91·23-s + 24-s + 25-s + 5.91·26-s + 27-s + 3.91·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.447·5-s + 0.408·6-s + 1.48·7-s + 0.353·8-s + 0.333·9-s − 0.316·10-s + 0.301·11-s + 0.288·12-s + 1.64·13-s + 1.04·14-s − 0.258·15-s + 0.250·16-s − 0.242·17-s + 0.235·18-s + 1.35·19-s − 0.223·20-s + 0.855·21-s + 0.213·22-s − 0.400·23-s + 0.204·24-s + 0.200·25-s + 1.16·26-s + 0.192·27-s + 0.740·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5610 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5610 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(5.122193642\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.122193642\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 17 | \( 1 + T \) |
good | 7 | \( 1 - 3.91T + 7T^{2} \) |
| 13 | \( 1 - 5.91T + 13T^{2} \) |
| 19 | \( 1 - 5.91T + 19T^{2} \) |
| 23 | \( 1 + 1.91T + 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 + 7.35T + 31T^{2} \) |
| 37 | \( 1 - 7.31T + 37T^{2} \) |
| 41 | \( 1 + 1.79T + 41T^{2} \) |
| 43 | \( 1 + 11.2T + 43T^{2} \) |
| 47 | \( 1 + 1.79T + 47T^{2} \) |
| 53 | \( 1 - 3.43T + 53T^{2} \) |
| 59 | \( 1 - 9.23T + 59T^{2} \) |
| 61 | \( 1 - 7.31T + 61T^{2} \) |
| 67 | \( 1 + 5.35T + 67T^{2} \) |
| 71 | \( 1 + 15.1T + 71T^{2} \) |
| 73 | \( 1 - 11.4T + 73T^{2} \) |
| 79 | \( 1 + 7.23T + 79T^{2} \) |
| 83 | \( 1 + 9.56T + 83T^{2} \) |
| 89 | \( 1 + 17.6T + 89T^{2} \) |
| 97 | \( 1 + 1.31T + 97T^{2} \) |
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
−8.203198055999801713426181830485, −7.44445813121992926361549975202, −6.82516961003895898792662753583, −5.77877440269518642567111818186, −5.22909678582982491761564118352, −4.29095267753841583704090525137, −3.80393638161371944847826660396, −3.02585566321338325947538062345, −1.83194560064604993920735697383, −1.21680773165388002741082183143,
1.21680773165388002741082183143, 1.83194560064604993920735697383, 3.02585566321338325947538062345, 3.80393638161371944847826660396, 4.29095267753841583704090525137, 5.22909678582982491761564118352, 5.77877440269518642567111818186, 6.82516961003895898792662753583, 7.44445813121992926361549975202, 8.203198055999801713426181830485