L(s) = 1 | − 2-s + 3-s + 4-s − 0.434·5-s − 6-s + 7-s − 8-s + 9-s + 0.434·10-s − 0.836·11-s + 12-s + 5.37·13-s − 14-s − 0.434·15-s + 16-s + 7.17·17-s − 18-s + 6.99·19-s − 0.434·20-s + 21-s + 0.836·22-s + 0.407·23-s − 24-s − 4.81·25-s − 5.37·26-s + 27-s + 28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.194·5-s − 0.408·6-s + 0.377·7-s − 0.353·8-s + 0.333·9-s + 0.137·10-s − 0.252·11-s + 0.288·12-s + 1.49·13-s − 0.267·14-s − 0.112·15-s + 0.250·16-s + 1.73·17-s − 0.235·18-s + 1.60·19-s − 0.0971·20-s + 0.218·21-s + 0.178·22-s + 0.0850·23-s − 0.204·24-s − 0.962·25-s − 1.05·26-s + 0.192·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8022 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8022 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.385681664\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.385681664\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 191 | \( 1 - T \) |
good | 5 | \( 1 + 0.434T + 5T^{2} \) |
| 11 | \( 1 + 0.836T + 11T^{2} \) |
| 13 | \( 1 - 5.37T + 13T^{2} \) |
| 17 | \( 1 - 7.17T + 17T^{2} \) |
| 19 | \( 1 - 6.99T + 19T^{2} \) |
| 23 | \( 1 - 0.407T + 23T^{2} \) |
| 29 | \( 1 - 2.11T + 29T^{2} \) |
| 31 | \( 1 + 3.77T + 31T^{2} \) |
| 37 | \( 1 - 3.36T + 37T^{2} \) |
| 41 | \( 1 - 5.35T + 41T^{2} \) |
| 43 | \( 1 - 4.89T + 43T^{2} \) |
| 47 | \( 1 + 3.06T + 47T^{2} \) |
| 53 | \( 1 + 1.54T + 53T^{2} \) |
| 59 | \( 1 + 4.93T + 59T^{2} \) |
| 61 | \( 1 + 2.97T + 61T^{2} \) |
| 67 | \( 1 - 11.3T + 67T^{2} \) |
| 71 | \( 1 - 3.86T + 71T^{2} \) |
| 73 | \( 1 - 8.15T + 73T^{2} \) |
| 79 | \( 1 - 3.13T + 79T^{2} \) |
| 83 | \( 1 + 13.6T + 83T^{2} \) |
| 89 | \( 1 - 10.1T + 89T^{2} \) |
| 97 | \( 1 + 6.77T + 97T^{2} \) |
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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
−7.77023802880925761286266741578, −7.62935647027419771362709308478, −6.58067954021440945762302948587, −5.75278369610294917962801144367, −5.23815853407391072322422136904, −3.98260853799421787037967042136, −3.43958283369516731040748346321, −2.68051257907849604679024443642, −1.50793019645522821475646023573, −0.931314738954447394093037178338,
0.931314738954447394093037178338, 1.50793019645522821475646023573, 2.68051257907849604679024443642, 3.43958283369516731040748346321, 3.98260853799421787037967042136, 5.23815853407391072322422136904, 5.75278369610294917962801144367, 6.58067954021440945762302948587, 7.62935647027419771362709308478, 7.77023802880925761286266741578