| L(s) = 1 | + 2.82·2-s − 3-s + 5.95·4-s − 1.22·5-s − 2.82·6-s + 7-s + 11.1·8-s + 9-s − 3.46·10-s + 4.62·11-s − 5.95·12-s + 1.56·13-s + 2.82·14-s + 1.22·15-s + 19.5·16-s + 5.88·17-s + 2.82·18-s − 6.41·19-s − 7.32·20-s − 21-s + 13.0·22-s + 3.54·23-s − 11.1·24-s − 3.48·25-s + 4.41·26-s − 27-s + 5.95·28-s + ⋯ |
| L(s) = 1 | + 1.99·2-s − 0.577·3-s + 2.97·4-s − 0.549·5-s − 1.15·6-s + 0.377·7-s + 3.94·8-s + 0.333·9-s − 1.09·10-s + 1.39·11-s − 1.71·12-s + 0.433·13-s + 0.753·14-s + 0.317·15-s + 4.89·16-s + 1.42·17-s + 0.664·18-s − 1.47·19-s − 1.63·20-s − 0.218·21-s + 2.77·22-s + 0.738·23-s − 2.27·24-s − 0.697·25-s + 0.865·26-s − 0.192·27-s + 1.12·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4767 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4767 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.129116752\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.129116752\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 227 | \( 1 + T \) |
| good | 2 | \( 1 - 2.82T + 2T^{2} \) |
| 5 | \( 1 + 1.22T + 5T^{2} \) |
| 11 | \( 1 - 4.62T + 11T^{2} \) |
| 13 | \( 1 - 1.56T + 13T^{2} \) |
| 17 | \( 1 - 5.88T + 17T^{2} \) |
| 19 | \( 1 + 6.41T + 19T^{2} \) |
| 23 | \( 1 - 3.54T + 23T^{2} \) |
| 29 | \( 1 + 5.86T + 29T^{2} \) |
| 31 | \( 1 + 3.39T + 31T^{2} \) |
| 37 | \( 1 + 10.7T + 37T^{2} \) |
| 41 | \( 1 - 1.73T + 41T^{2} \) |
| 43 | \( 1 - 9.24T + 43T^{2} \) |
| 47 | \( 1 + 2.36T + 47T^{2} \) |
| 53 | \( 1 + 9.95T + 53T^{2} \) |
| 59 | \( 1 - 8.49T + 59T^{2} \) |
| 61 | \( 1 + 13.1T + 61T^{2} \) |
| 67 | \( 1 + 0.520T + 67T^{2} \) |
| 71 | \( 1 - 5.93T + 71T^{2} \) |
| 73 | \( 1 - 5.57T + 73T^{2} \) |
| 79 | \( 1 - 9.92T + 79T^{2} \) |
| 83 | \( 1 + 2.75T + 83T^{2} \) |
| 89 | \( 1 + 0.433T + 89T^{2} \) |
| 97 | \( 1 - 12.1T + 97T^{2} \) |
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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
−7.78206803530036637986394188763, −7.31744569997289288003947263357, −6.47228484740707099487328992458, −6.01636187033370925496748326570, −5.26726066724897317852342555438, −4.55526359451836152285351341210, −3.73188923682652537143181583418, −3.52852064228665952577610860813, −2.06693688642671915623278154567, −1.28537614114412229340500120686,
1.28537614114412229340500120686, 2.06693688642671915623278154567, 3.52852064228665952577610860813, 3.73188923682652537143181583418, 4.55526359451836152285351341210, 5.26726066724897317852342555438, 6.01636187033370925496748326570, 6.47228484740707099487328992458, 7.31744569997289288003947263357, 7.78206803530036637986394188763
