| L(s) = 1 | − 1.95·2-s + 3-s + 1.83·4-s − 2.04·5-s − 1.95·6-s + 0.328·7-s + 0.324·8-s + 9-s + 4.01·10-s + 6.23·11-s + 1.83·12-s − 2.39·13-s − 0.644·14-s − 2.04·15-s − 4.30·16-s + 4.05·17-s − 1.95·18-s + 5.45·19-s − 3.76·20-s + 0.328·21-s − 12.2·22-s − 5.21·23-s + 0.324·24-s − 0.798·25-s + 4.69·26-s + 27-s + 0.603·28-s + ⋯ |
| L(s) = 1 | − 1.38·2-s + 0.577·3-s + 0.917·4-s − 0.916·5-s − 0.799·6-s + 0.124·7-s + 0.114·8-s + 0.333·9-s + 1.26·10-s + 1.87·11-s + 0.529·12-s − 0.664·13-s − 0.172·14-s − 0.529·15-s − 1.07·16-s + 0.983·17-s − 0.461·18-s + 1.25·19-s − 0.840·20-s + 0.0717·21-s − 2.60·22-s − 1.08·23-s + 0.0661·24-s − 0.159·25-s + 0.920·26-s + 0.192·27-s + 0.114·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2031 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2031 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9970000171\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9970000171\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 677 | \( 1 + T \) |
| good | 2 | \( 1 + 1.95T + 2T^{2} \) |
| 5 | \( 1 + 2.04T + 5T^{2} \) |
| 7 | \( 1 - 0.328T + 7T^{2} \) |
| 11 | \( 1 - 6.23T + 11T^{2} \) |
| 13 | \( 1 + 2.39T + 13T^{2} \) |
| 17 | \( 1 - 4.05T + 17T^{2} \) |
| 19 | \( 1 - 5.45T + 19T^{2} \) |
| 23 | \( 1 + 5.21T + 23T^{2} \) |
| 29 | \( 1 - 7.79T + 29T^{2} \) |
| 31 | \( 1 - 4.92T + 31T^{2} \) |
| 37 | \( 1 + 8.20T + 37T^{2} \) |
| 41 | \( 1 + 5.43T + 41T^{2} \) |
| 43 | \( 1 - 3.93T + 43T^{2} \) |
| 47 | \( 1 + 7.42T + 47T^{2} \) |
| 53 | \( 1 - 0.108T + 53T^{2} \) |
| 59 | \( 1 - 11.5T + 59T^{2} \) |
| 61 | \( 1 + 5.74T + 61T^{2} \) |
| 67 | \( 1 + 1.24T + 67T^{2} \) |
| 71 | \( 1 - 14.4T + 71T^{2} \) |
| 73 | \( 1 - 6.42T + 73T^{2} \) |
| 79 | \( 1 + 5.60T + 79T^{2} \) |
| 83 | \( 1 - 9.84T + 83T^{2} \) |
| 89 | \( 1 - 8.44T + 89T^{2} \) |
| 97 | \( 1 + 8.32T + 97T^{2} \) |
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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
−9.131431510964603607006892737977, −8.256914501221839137061828867923, −7.950742157556071765185212440440, −7.10055968662257091661573554900, −6.48410000245131169193618850926, −4.97342057637573402649560478620, −4.02328013997785235399630486572, −3.24371262208005251268396312744, −1.79827256196521216005699843227, −0.843235585868650809478688335428,
0.843235585868650809478688335428, 1.79827256196521216005699843227, 3.24371262208005251268396312744, 4.02328013997785235399630486572, 4.97342057637573402649560478620, 6.48410000245131169193618850926, 7.10055968662257091661573554900, 7.950742157556071765185212440440, 8.256914501221839137061828867923, 9.131431510964603607006892737977
