| L(s) = 1 | − 0.960·2-s + 3-s − 1.07·4-s − 2.82·5-s − 0.960·6-s − 0.386·7-s + 2.95·8-s + 9-s + 2.71·10-s − 1.78·11-s − 1.07·12-s + 1.64·13-s + 0.371·14-s − 2.82·15-s − 0.682·16-s − 17-s − 0.960·18-s − 0.409·19-s + 3.04·20-s − 0.386·21-s + 1.71·22-s + 6.35·23-s + 2.95·24-s + 2.97·25-s − 1.57·26-s + 27-s + 0.416·28-s + ⋯ |
| L(s) = 1 | − 0.679·2-s + 0.577·3-s − 0.538·4-s − 1.26·5-s − 0.392·6-s − 0.146·7-s + 1.04·8-s + 0.333·9-s + 0.857·10-s − 0.538·11-s − 0.311·12-s + 0.455·13-s + 0.0992·14-s − 0.729·15-s − 0.170·16-s − 0.242·17-s − 0.226·18-s − 0.0939·19-s + 0.680·20-s − 0.0843·21-s + 0.365·22-s + 1.32·23-s + 0.603·24-s + 0.595·25-s − 0.309·26-s + 0.192·27-s + 0.0787·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7496017541\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7496017541\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| 79 | \( 1 + T \) |
| good | 2 | \( 1 + 0.960T + 2T^{2} \) |
| 5 | \( 1 + 2.82T + 5T^{2} \) |
| 7 | \( 1 + 0.386T + 7T^{2} \) |
| 11 | \( 1 + 1.78T + 11T^{2} \) |
| 13 | \( 1 - 1.64T + 13T^{2} \) |
| 19 | \( 1 + 0.409T + 19T^{2} \) |
| 23 | \( 1 - 6.35T + 23T^{2} \) |
| 29 | \( 1 - 3.35T + 29T^{2} \) |
| 31 | \( 1 + 5.19T + 31T^{2} \) |
| 37 | \( 1 + 9.22T + 37T^{2} \) |
| 41 | \( 1 - 0.201T + 41T^{2} \) |
| 43 | \( 1 - 3.48T + 43T^{2} \) |
| 47 | \( 1 + 13.6T + 47T^{2} \) |
| 53 | \( 1 + 2.52T + 53T^{2} \) |
| 59 | \( 1 - 8.70T + 59T^{2} \) |
| 61 | \( 1 + 5.26T + 61T^{2} \) |
| 67 | \( 1 + 8.65T + 67T^{2} \) |
| 71 | \( 1 - 6.61T + 71T^{2} \) |
| 73 | \( 1 - 8.99T + 73T^{2} \) |
| 83 | \( 1 - 7.46T + 83T^{2} \) |
| 89 | \( 1 - 15.5T + 89T^{2} \) |
| 97 | \( 1 - 16.3T + 97T^{2} \) |
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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
−8.426197042250328519047101022213, −7.917318381607610650952888371753, −7.30356481272900959449958509278, −6.56989080224684774498024328294, −5.18318806533741370897217707501, −4.62957330995693385027633680015, −3.69359817055752054713177781079, −3.17501035451906612346384506987, −1.77541926242229028841718208120, −0.54007558318261560701240613273,
0.54007558318261560701240613273, 1.77541926242229028841718208120, 3.17501035451906612346384506987, 3.69359817055752054713177781079, 4.62957330995693385027633680015, 5.18318806533741370897217707501, 6.56989080224684774498024328294, 7.30356481272900959449958509278, 7.917318381607610650952888371753, 8.426197042250328519047101022213
