| L(s) = 1 | + 2-s − 3-s + 4-s + 3.56·5-s − 6-s − 3.12·7-s + 8-s + 9-s + 3.56·10-s + 1.56·11-s − 12-s + 3.56·13-s − 3.12·14-s − 3.56·15-s + 16-s − 6.68·17-s + 18-s + 1.56·19-s + 3.56·20-s + 3.12·21-s + 1.56·22-s − 8·23-s − 24-s + 7.68·25-s + 3.56·26-s − 27-s − 3.12·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s + 1.59·5-s − 0.408·6-s − 1.18·7-s + 0.353·8-s + 0.333·9-s + 1.12·10-s + 0.470·11-s − 0.288·12-s + 0.987·13-s − 0.834·14-s − 0.919·15-s + 0.250·16-s − 1.62·17-s + 0.235·18-s + 0.358·19-s + 0.796·20-s + 0.681·21-s + 0.332·22-s − 1.66·23-s − 0.204·24-s + 1.53·25-s + 0.698·26-s − 0.192·27-s − 0.590·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 186 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 186 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.675381320\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.675381320\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 31 | \( 1 + T \) |
| good | 5 | \( 1 - 3.56T + 5T^{2} \) |
| 7 | \( 1 + 3.12T + 7T^{2} \) |
| 11 | \( 1 - 1.56T + 11T^{2} \) |
| 13 | \( 1 - 3.56T + 13T^{2} \) |
| 17 | \( 1 + 6.68T + 17T^{2} \) |
| 19 | \( 1 - 1.56T + 19T^{2} \) |
| 23 | \( 1 + 8T + 23T^{2} \) |
| 29 | \( 1 - 1.12T + 29T^{2} \) |
| 37 | \( 1 + 8.24T + 37T^{2} \) |
| 41 | \( 1 + 12.2T + 41T^{2} \) |
| 43 | \( 1 + 0.876T + 43T^{2} \) |
| 47 | \( 1 - 8.68T + 47T^{2} \) |
| 53 | \( 1 + 2T + 53T^{2} \) |
| 59 | \( 1 - 7.12T + 59T^{2} \) |
| 61 | \( 1 - 3.56T + 61T^{2} \) |
| 67 | \( 1 - 12.6T + 67T^{2} \) |
| 71 | \( 1 + 5.56T + 71T^{2} \) |
| 73 | \( 1 - 10T + 73T^{2} \) |
| 79 | \( 1 - 8.68T + 79T^{2} \) |
| 83 | \( 1 + 7.80T + 83T^{2} \) |
| 89 | \( 1 - 16.2T + 89T^{2} \) |
| 97 | \( 1 - 5.80T + 97T^{2} \) |
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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
−12.82970242883536565766697726296, −11.77635719893748808657083014272, −10.58074293327207340388459008405, −9.848792134156071187791953251953, −8.813422363596946949301763839220, −6.65478381999053354976868805730, −6.33537835797121285546506423012, −5.32146829320858463202368391376, −3.74481057082569996274433617289, −2.02882643892120175759435607569,
2.02882643892120175759435607569, 3.74481057082569996274433617289, 5.32146829320858463202368391376, 6.33537835797121285546506423012, 6.65478381999053354976868805730, 8.813422363596946949301763839220, 9.848792134156071187791953251953, 10.58074293327207340388459008405, 11.77635719893748808657083014272, 12.82970242883536565766697726296
