| L(s) = 1 | + 2-s + 3-s + 4-s − 0.880·5-s + 6-s − 1.31·7-s + 8-s + 9-s − 0.880·10-s − 0.806·11-s + 12-s + 0.854·13-s − 1.31·14-s − 0.880·15-s + 16-s + 17-s + 18-s − 3.32·19-s − 0.880·20-s − 1.31·21-s − 0.806·22-s − 8.04·23-s + 24-s − 4.22·25-s + 0.854·26-s + 27-s − 1.31·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.393·5-s + 0.408·6-s − 0.496·7-s + 0.353·8-s + 0.333·9-s − 0.278·10-s − 0.243·11-s + 0.288·12-s + 0.236·13-s − 0.350·14-s − 0.227·15-s + 0.250·16-s + 0.242·17-s + 0.235·18-s − 0.762·19-s − 0.196·20-s − 0.286·21-s − 0.171·22-s − 1.67·23-s + 0.204·24-s − 0.844·25-s + 0.167·26-s + 0.192·27-s − 0.248·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| 59 | \( 1 - T \) |
| good | 5 | \( 1 + 0.880T + 5T^{2} \) |
| 7 | \( 1 + 1.31T + 7T^{2} \) |
| 11 | \( 1 + 0.806T + 11T^{2} \) |
| 13 | \( 1 - 0.854T + 13T^{2} \) |
| 19 | \( 1 + 3.32T + 19T^{2} \) |
| 23 | \( 1 + 8.04T + 23T^{2} \) |
| 29 | \( 1 - 4.13T + 29T^{2} \) |
| 31 | \( 1 + 3.08T + 31T^{2} \) |
| 37 | \( 1 + 10.6T + 37T^{2} \) |
| 41 | \( 1 - 2.28T + 41T^{2} \) |
| 43 | \( 1 - 1.19T + 43T^{2} \) |
| 47 | \( 1 - 1.81T + 47T^{2} \) |
| 53 | \( 1 - 5.11T + 53T^{2} \) |
| 61 | \( 1 + 4.72T + 61T^{2} \) |
| 67 | \( 1 + 5.94T + 67T^{2} \) |
| 71 | \( 1 + 2.49T + 71T^{2} \) |
| 73 | \( 1 + 3.28T + 73T^{2} \) |
| 79 | \( 1 - 10.9T + 79T^{2} \) |
| 83 | \( 1 + 17.5T + 83T^{2} \) |
| 89 | \( 1 + 13.3T + 89T^{2} \) |
| 97 | \( 1 - 14.6T + 97T^{2} \) |
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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
−7.68945076180811233608427296995, −7.02279748041382042582358709445, −6.21714159769921386435355599128, −5.65509889057975447039072016208, −4.63027323617447782063062517159, −3.93322139837896413027093307884, −3.40337901466880931151130881975, −2.47724459361238833055537054952, −1.65597636120130251981841318282, 0,
1.65597636120130251981841318282, 2.47724459361238833055537054952, 3.40337901466880931151130881975, 3.93322139837896413027093307884, 4.63027323617447782063062517159, 5.65509889057975447039072016208, 6.21714159769921386435355599128, 7.02279748041382042582358709445, 7.68945076180811233608427296995
